2000 character limit reached
Hierarchical cycle-tree packing model for $K$-core attack problem
Published 2 Mar 2023 in cond-mat.dis-nn, cond-mat.stat-mech, cs.CY, and physics.soc-ph | (2303.01007v2)
Abstract: The $K$-core of a graph is the unique maximum subgraph within which each vertex connects to $K$ or more other vertices. The optimal $K$-core attack problem asks to delete the minimum number of vertices from the $K$-core to induce its complete collapse. A hierarchical cycle-tree packing model is introduced here for this challenging combinatorial optimization problem. We convert the temporally long-range correlated $K$-core pruning dynamics into locally tree-like static patterns and analyze this model through the replica-symmetric cavity method of statistical physics. A set of coarse-grained belief propagation equations are derived to predict single vertex marginal probabilities efficiently. The associated hierarchical cycle-tree guided attack ({\tt hCTGA}) algorithm is able to construct nearly optimal attack solutions for regular random graphs and Erd\"os-R\'enyi random graphs. Our cycle-tree packing model may also be helpful for constructing optimal initial conditions for other irreversible dynamical processes on sparse random graphs.
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Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. 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URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. 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American Journal of Sociology 83(6), 1420–1443 (1978). 10.1086/226707. URL https://www.journals.uchicago.edu/doi/abs/10.1086/226707 (4) S.B. Seidman, Network structure and minimum degree. Social Networks 5(3), 269–287 (1983). 10.1016/0378-8733(83)90028-x. URL https://www.sciencedirect.com/science/article/abs/pii/037887338390028X?via%3Dihub (5) Y.X. Kong, G.Y. Shi, R.J. Wu, Y.C. Zhang, K-core: Theories and applications. Physics Reports 832, 1–32 (2019). 10.1016/j.physrep.2019.10.004. URL https://www.sciencedirect.com/science/article/pii/S037015731930328X (6) G.H. Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. 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Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. 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URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. 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Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G.H. Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. 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Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. 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Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. 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Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. 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Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. 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URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. 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URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.B. Seidman, Network structure and minimum degree. Social Networks 5(3), 269–287 (1983). 10.1016/0378-8733(83)90028-x. URL https://www.sciencedirect.com/science/article/abs/pii/037887338390028X?via%3Dihub (5) Y.X. Kong, G.Y. Shi, R.J. Wu, Y.C. Zhang, K-core: Theories and applications. Physics Reports 832, 1–32 (2019). 10.1016/j.physrep.2019.10.004. URL https://www.sciencedirect.com/science/article/pii/S037015731930328X (6) G.H. Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.X. Kong, G.Y. Shi, R.J. Wu, Y.C. Zhang, K-core: Theories and applications. Physics Reports 832, 1–32 (2019). 10.1016/j.physrep.2019.10.004. URL https://www.sciencedirect.com/science/article/pii/S037015731930328X (6) G.H. Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. 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URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. 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URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. 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Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. 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Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. 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URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. 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Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. 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URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. 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Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G.H. Fredrickson, H.C. Andersen, Kinetic Ising model of the glass transition. Physical Review Letters 53(13), 1244–1247 (1984). 10.1103/PhysRevLett.53.1244. URL https://link.aps.org/doi/10.1103/PhysRevLett.53.1244 (7) M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. 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URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. 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Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. 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URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. 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Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. 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URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. 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Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. 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Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. 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Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Sellitto, G. Biroli, C. Toninelli, Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhysics Letters 69(4), 496 (2005). 10.1209/epl/i2004-10372-5. URL https://dx.doi.org/10.1209/epl/i2004-10372-5 (8) T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, Fate of the hybrid transition of bootstrap percolation in physical dimension. Physical Review Letters 122(10), 108301 (2019). 10.1103/PhysRevLett.122.108301. URL https://link.aps.org/doi/10.1103/PhysRevLett.122.108301 (9) T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. 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Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. 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URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. 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Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. 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URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. 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Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. URL https://ui.adsabs.harvard.edu/abs/2022arXiv221205132P (11) M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. 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Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf T. Rizzo, T. Voigtmann, Solvable models of supercooled liquids in three dimensions. Physical Review Letters 124(19), 195,501 (2020). 10.1103/PhysRevLett.124.195501. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.195501 (10) G. Perrupato, T. Rizzo, Exact dynamical equations for kinetically-constrained-models. e-print arXiv:2212.05132 (2022). 10.48550/arXiv.2212.05132. 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URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Altaf-Ul-Amine, K. Nishikata, T. Korna, T. Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, T. Oshima, H. Mori, S. Kanaya, Prediction of protein functions based on K-cores of protein-protein interaction networks and amino acid sequences. Genome Informatics 14, 498–499 (2003). 10.11234/gi1990.14.498. URL https://www.jstage.jst.go.jp/article/gi1990/14/0/14_0_498/_article (12) S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. 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URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. 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Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. 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Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. 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URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Wuchty, E. Almaas, Peeling the yeast protein network. PROTEOMICS 5(2), 444–449 (2005). 10.1002/pmic.200400962. URL https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/pmic.200400962 (13) F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. 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URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. 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The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. 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Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Arese Lucini, G. Del Ferraro, M. Sigman, H.A. Makse, How the brain transitions from conscious to subliminal perception. Neuroscience 411, 280–290 (2019). 10.1016/j.neuroscience.2019.03.047. URL https://www.sciencedirect.com/science/article/pii/S0306452219302052 (14) W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf W.C. Stanford, P.J. Mucha, E. Dayan, A robust core architecture of functional brain networks supports topological resilience and cognitive performance in middle- and old-aged adults. Proceedings of the National Academy of Sciences 119(44), e2203682,119 (2022). 10.1073/pnas.2203682119. URL https://www.pnas.org/doi/abs/10.1073/pnas.2203682119 (15) J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J. GarcÃa-Algarra, J.M. Pastor, J.M. Iriondo, J. Galeano, Ranking of critical species to preserve the functionality of mutualistic networks using the K-core decomposition. PeerJ 5, e3321 (2017). 10.7717/peerj.3321. URL https://doi.org/10.7717/peerj.3321 (16) M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Kitsak, L.K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H.E. Stanley, H.A. Makse, Identification of influential spreaders in complex networks. Nature Physics 6(11), 888–893 (2010). 10.1038/nphys1746. URL https://doi.org/10.1038/nphys1746 (17) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core organization of complex networks. Physical Review Letters 96(4), 040601 (2006). 10.1103/PhysRevLett.96.040601. URL https://link.aps.org/doi/10.1103/PhysRevLett.96.040601 (18) S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, K-core architecture and K-core percolation on complex networks. Physica D: Nonlinear Phenomena 224(1), 7–19 (2006). 10.1016/j.physd.2006.09.027. URL https://www.sciencedirect.com/science/article/pii/S0167278906003617 (19) N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. 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Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf N. Azimi-Tafreshi, J. Gómez-Gardeñes, S.N. Dorogovtsev, K-core percolation on multiplex networks. Physical Review E 90(3), 032,816 (2014). 10.1103/PhysRevE.90.032816. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032816 (20) G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf G. J. Baxter, S. N. Dorogovtsev, K.-E. Lee, J. F. F. Mendes, A. V. Goltsev, Critical dynamics of the K-core pruning process. Physical Review X 5(3), 031017 (2015). 10.1103/PhysRevX.5.031017. URL https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.031017 (21) R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. 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URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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(25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.J. Wu, Y.X. Kong, Z. Di, Y.C. Zhang, G.Y. Shi, Analytical solution to the K-core pruning process. Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Physica A 608, 128260 (2022). 10.1016/j.physa.2022.128260. URL https://www.sciencedirect.com/science/article/pii/S0378437122008184 (22) J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Y.Y. Liu, Inducing effect on the percolation transition in complex networks. Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Nature Commun. 4, 2412 (2013). doi: 10.1038/ncomms3412 (23) J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, Generalized K-core pruning process on directed networks. Journal of Statistical Mechanics: Theory and Experiment 2017, 063407 (2017). 10.1088/1742-5468/aa71e0. URL https://iopscience.iop.org/article/10.1088/1742-5468/aa71e0 (24) S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. 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URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S.N. Wang, L. Cheng, H.J. Zhou, Vulnerability and resilience of social engagement: Equilibrium theory. Europhys. Lett. 132, 60006 (2020). https://doi.org/10.1209/0295-5075/132/60006. (25) A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Guggiola, G. Semerjian, Minimal contagious sets in random regular graphs. Journal of Statistical Physics 158(2), 300–358 (2015). 10.1007/s10955-014-1136-2. URL https://doi.org/10.1007/s10955-014-1136-2 (26) C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf C. Schmidt, H.D. Pfister, L. Zdeborová, Minimal sets to destroy the K-core in random networks. Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. 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Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Phys Rev E 99(2-1), 022,310 (2019). 10.1103/PhysRevE.99.022310. URL https://www.ncbi.nlm.nih.gov/pubmed/30934241 (27) H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- H.J. Zhou, Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm. Science China Physics, Mechanics & Astronomy 65(3), 230,511 (2022). 10.1007/s11433-021-1845-6. URL https://link.springer.com/article/10.1007/s11433-021-1845-6 (28) R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R. Ma, Y. Hu, J.H. Zhao, Random node reinforcement and k-core structure of complex networks. Chaos Solitons & Fractals 173, 113706 (2023). 10.1016/j.chaos.2023.113706 (29) R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. 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Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf R.M. Karp, Reducibility among Combinatorial Problems, in Complexity of Computer Computations (Springer US, Boston, MA, 1972), pp. 85–103. 10.1007/978-1-4684-2001-2_9. URL https://doi.org/10.1007/978-1-4684-2001-2_9 (30) F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. 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Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F.V. Fomin, S. Gaspers, A.V. Pyatkin, I. Razgon, On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Algorithmica 52, 293–307 (2008). 10.1007/s00453-007-9152-0 (31) S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Bau, N.C. Wormald, S. Zhou, Decycling numbers of random regular graphs. Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Random Structures & Algorithms 21(3-4), 397–413 (2002). 10.1002/rsa.10069. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.10069 (32) H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- H.J. Zhou, Spin glass approach to the feedback vertex set problem. The European Physical Journal B 86(11), 455 (2013). 10.1140/epjb/e2013-40690-1. URL https://doi.org/10.1140/epjb/e2013-40690-1 (33) H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- H.J. Zhou, A spin glass approach to the directed feedback vertex set problem. Journal of Statistical Mechanics: Theory and Experiment 2016(7), 073303 (2016). 10.1088/1742-5468/2016/07/073303. URL http://dx.doi.org/10.1088/1742-5468/2016/07/073303 (34) A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Braunstein, L. Dall’Asta, G. Semerjian, L. Zdeborová, Network dismantling. Proceedings of the National Academy of Sciences 113(44), 12368–12373 (2016). 10.1073/pnas.1605083113. URL https://dx.doi.org/10.1073/pnas.1605083113 (35) L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf L. Zdeborová, P. Zhang, H.J. Zhou, Fast and simple decycling and dismantling of networks. Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. 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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. 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URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. 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URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Scientific Reports 6(1), 37954 (2016). 10.1038/srep37954. URL https://doi.org/10.1038/srep37954 (36) S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf S. Pei, X. Teng, J. Shaman, F. Morone, H.A. Makse, Efficient collective influence maximization in cascading processes with first-order transitions. Scientific Reports 7(1), 45240 (2017). 10.1038/srep45240. URL https://doi.org/10.1038/srep45240 (37) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Optimizing spread dynamics on graphs by message passing. Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Journal of Statistical Mechanics: Theory and Experiment 2013(09), P09011 (2013). 10.1088/1742-5468/2013/09/p09011. URL http://dx.doi.org/10.1088/1742-5468/2013/09/P09011 (38) F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. 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URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Altarelli, A. Braunstein, L. Dall’Asta, R. Zecchina, Large deviations of cascade processes on graphs. Physical Review E 87(6), 062115 (2013). 10.1103/physreve.87.062115. URL https://dx.doi.org/10.1103/physreve.87.062115 (39) D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. 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Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf D. Reichman, New bounds for contagious sets. Discrete Mathematics 312(10), 1812–1814 (2012). 10.1016/j.disc.2012.01.016. URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. 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URL https://www.sciencedirect.com/science/article/pii/S0012365X12000301 (40) A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf A. Coja-Oghlan, U. Feige, M. Krivelevich, D. Reichman, Contagious Sets in Expanders, in Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2014), pp. 1953–1987. 10.1137/1.9781611973730.131. URL https://doi.org/10.1137/1.9781611973730.131 (41) P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. 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Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf P.A. Dreyer, F.S. Roberts, Irreversible K-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion. Discrete Applied Mathematics 157(7), 1615–1627 (2009). 10.1016/j.dam.2008.09.012. URL https://www.sciencedirect.com/science/article/pii/S0166218X08004393 (42) F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf F. Zhang, Y. Zhang, L. Qin, W. Zhang, X. Lin, Finding critical users for social network engagement: The collapsed k-core problem. Proceedings of the AAAI Conference on Artificial Intelligence 31(1), 245–251 (2017). 10.1609/aaai.v31i1.10482. URL https://ojs.aaai.org/index.php/AAAI/article/view/10482 (43) J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. URL https://arxiv.org/abs/1605.09257 (44) M. Mézard, A. Montanari, Information, Physics, and Computation (Oxford University Press, 2009). 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf J.H. Zhao, H.J. Zhou, Optimal disruption of complex networks. e-print arXiv:1605.09257 (2016). 10.48550/arXiv.1605.09257. 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Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. 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URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. 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- 10.1093/acprof:oso/9780198570837.001.0001. URL https://academic.oup.com/book/6337 (45) H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf H. Zhou, Spin Glass and Message Passing (Science Press, Beijing, 2015) (46) Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Y. Habibulla, H.J. Zhou, Minimum connected dominating set and backbone of a random graph. e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- e-print arXiv:2310.15980 (2023). 10.48550/arXiv.2310.15980. (47) Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf Y.Z. Xu, H.J. Zhou, Optimal segmentation of directed graph and the minimum number of feedback arcs. Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- Journal of Statistical Physics 169(1), 187–202 (2017). 10.1007/s10955-017-1860-5. URL https://link.springer.com/article/10.1007/s10955-017-1860-5 (48) B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf B. Zhou, Y. Lv, J. Wang, J. Zhang, Q. Xuan, COREATTACK: Breaking Up the Core Structure of Graphs. arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- arXiv e-prints (2021). 10.48550/arXiv.2111.15276. URL https://ui.adsabs.harvard.edu/abs/2021arXiv211115276Z (49) X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf X.J. Wang, J. Jiang, U. Pereira-Obilinovic, Bifurcation in space: Emergence of function modularity in the neocortex. e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
- e-print bioRxiv:2023.06.04.543639 (2023). 10.1101/2023.06.04.543639. URL https://www.biorxiv.org/content/10.1101/2023.06.04.543639v1.full.pdf
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