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On Metzler positive systems on hypergraphs (2401.03652v3)

Published 8 Jan 2024 in eess.SY and cs.SY

Abstract: In graph-theoretical terms, an edge in a graph connects two vertices while a hyperedge of a hypergraph connects any more than one vertices. If the hypergraph's hyperedges further connect the same number of vertices, it is said to be uniform. In algebraic graph theory, a graph can be characterized by an adjacency matrix, and similarly, a uniform hypergraph can be characterized by an adjacency tensor. This similarity enables us to extend existing tools of matrix analysis for studying dynamical systems evolving on graphs to the study of a class of polynomial dynamical systems evolving on hypergraphs utilizing the properties of tensors. To be more precise, in this paper, we first extend the concept of a Metzler matrix to a Metzler tensor and then describe some useful properties of such tensors. Next, we focus on positive systems on hypergraphs associated with Metzler tensors. More importantly, we design control laws to stabilize the origin of this class of Metzler positive systems on hypergraphs. In the end, we apply our findings to two classic dynamical systems: a higher-order Lotka-Volterra population dynamics system and a higher-order SIS epidemic dynamic process. The corresponding novel stability results are accompanied by ample numerical examples.

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References (46)
  1. G. Craciun, “Polynomial dynamical systems, reaction networks, and toric differential inclusions,” SIAM Journal on Applied Algebra and Geometry, vol. 3, no. 1, pp. 87–106, 2019.
  2. P. Donnell and M. Banaji, “Local and global stability of equilibria for a class of chemical reaction networks,” SIAM Journal on Applied Dynamical Systems, vol. 12, no. 2, pp. 899–920, 2013.
  3. G. Craciun, Y. Tang, and M. Feinberg, “Understanding bistability in complex enzyme-driven reaction networks,” Proceedings of the National Academy of Sciences, vol. 103, no. 23, pp. 8697–8702, 2006.
  4. D. Angeli, “A tutorial on chemical reaction networks dynamics,” in 2009 European Control Conference (ECC).   IEEE, 2009, pp. 649–657.
  5. W. Ji and S. Deng, “Autonomous discovery of unknown reaction pathways from data by chemical reaction neural network,” The Journal of Physical Chemistry A, vol. 125, no. 4, pp. 1082–1092, 2021.
  6. Y. Takeuchi, N. Adachi, and H. Tokumaru, “The stability of generalized volterra equations,” Journal of Mathematical Analysis and Applications, vol. 62, no. 3, pp. 453–473, 1978.
  7. B. Goh, “Global stability in two species interactions,” Journal of Mathematical Biology, vol. 3, no. 3, pp. 313–318, 1976.
  8. ——, “Stability in models of mutualism,” The American Naturalist, vol. 113, no. 2, pp. 261–275, 1979.
  9. F. Liu, C. Shaoxuan, X. Li, and M. Buss, “On the stability of the endemic equilibrium of a discrete-time networked epidemic model,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2576–2581, 2020.
  10. S. Cui, F. Liu, H. Jardón-Kojakhmetov, and M. Cao, “Discrete-time layered-network epidemics model with time-varying transition rates and multiple resources,” arXiv preprint arXiv:2206.07425, 2022.
  11. P. E. Paré, J. Liu, C. L. Beck, B. E. Kirwan, and T. Başar, “Analysis, estimation, and validation of discrete-time epidemic processes,” IEEE Transactions on Control Systems Technology, vol. 28, no. 1, pp. 79–93, 2018.
  12. J. Liu, P. E. Paré, A. Nedić, C. Y. Tang, C. L. Beck, and T. Başar, “Analysis and control of a continuous-time bi-virus model,” IEEE Transactions on Automatic Control, vol. 64, no. 12, pp. 4891–4906, 2019.
  13. C. Zhang, S. Gracy, T. Basar, and P. E. Pare, “Analysis, control, and state estimation for the networked competitive multi-virus sir model,” arXiv preprint arXiv:2305.09068, 2023.
  14. C. Zhang, H. Leung, B. A. Butler, and P. E. Paré, “Estimation and distributed eradication of sir epidemics on networks,” IEEE Transactions on Control of Network Systems, 2023.
  15. F. Liu and M. Buss, “Node-based sirs model on heterogeneous networks: Analysis and control,” in 2016 American Control Conference (ACC).   IEEE, 2016, pp. 2852–2857.
  16. F. Liu, Z. Zhang, and M. Buss, “Optimal filtering and control of network information epidemics,” at-Automatisierungstechnik, vol. 69, no. 2, pp. 122–130, 2021.
  17. M. M. Mayfield and D. B. Stouffer, “Higher-order interactions capture unexplained complexity in diverse communities,” Nature ecology & evolution, vol. 1, no. 3, pp. 1–7, 2017.
  18. A. D. Letten and D. B. Stouffer, “The mechanistic basis for higher-order interactions and non-additivity in competitive communities,” Ecology letters, vol. 22, no. 3, pp. 423–436, 2019.
  19. I. Iacopini, G. Petri, A. Barrat, and V. Latora, “Simplicial models of social contagion,” Nature communications, vol. 10, no. 1, p. 2485, 2019.
  20. P. Cisneros-Velarde and F. Bullo, “Multigroup sis epidemics with simplicial and higher order interactions,” IEEE Transactions on Control of Network Systems, vol. 9, no. 2, pp. 695–705, 2021.
  21. S. Cui, F. Liu, H. Jardón-Kojakhmetov, and M. Cao, “General sis diffusion process with indirect spreading pathways on a hypergraph,” arXiv preprint arXiv:2306.00619, 2023.
  22. G. St-Onge, H. Sun, A. Allard, L. Hébert-Dufresne, and G. Bianconi, “Universal nonlinear infection kernel from heterogeneous exposure on higher-order networks,” Physical review letters, vol. 127, no. 15, p. 158301, 2021.
  23. W. Li, X. Xue, L. Pan, T. Lin, and W. Wang, “Competing spreading dynamics in simplicial complex,” Applied Mathematics and Computation, vol. 412, p. 126595, 2022.
  24. C. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, “What are higher-order networks?” SIAM Review, vol. 65, no. 3, pp. 686–731, 2023.
  25. G. Gallo, G. Longo, S. Pallottino, and S. Nguyen, “Directed hypergraphs and applications,” Discrete applied mathematics, vol. 42, no. 2-3, pp. 177–201, 1993.
  26. K. Chang, L. Qi, and T. Zhang, “A survey on the spectral theory of nonnegative tensors,” Numerical Linear Algebra with Applications, vol. 20, no. 6, pp. 891–912, 2013.
  27. K.-C. Chang, K. Pearson, and T. Zhang, “Perron-frobenius theorem for nonnegative tensors,” Communications in Mathematical Sciences, vol. 6, no. 2, pp. 507–520, 2008.
  28. Y. Yang and Q. Yang, “Further results for perron–frobenius theorem for nonnegative tensors,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 5, pp. 2517–2530, 2010.
  29. Q. Yang and Y. Yang, “Further results for perron–frobenius theorem for nonnegative tensors ii,” SIAM Journal on Matrix Analysis and Applications, vol. 32, no. 4, pp. 1236–1250, 2011.
  30. E. Okyere, A. Bousbaine, G. T. Poyi, A. K. Joseph, and J. M. Andrade, “Lqr controller design for quad-rotor helicopters,” The Journal of Engineering, vol. 2019, no. 17, pp. 4003–4007, 2019.
  31. A. Rantzer, “Distributed control of positive systems,” in 2011 50th IEEE conference on decision and control and European control conference.   IEEE, 2011, pp. 6608–6611.
  32. L. Zhang, L. Qi, and G. Zhou, “M-tensors and some applications,” SIAM Journal on Matrix Analysis and Applications, vol. 35, no. 2, pp. 437–452, 2014.
  33. W. Ding, L. Qi, and Y. Wei, “M-tensors and nonsingular m-tensors,” Linear Algebra and Its Applications, vol. 439, no. 10, pp. 3264–3278, 2013.
  34. W. Ding and Y. Wei, “Solving multi-linear systems with m-tensors,” Journal of Scientific Computing, vol. 68, no. 2, pp. 689–715, 2016.
  35. L. Qi, “Eigenvalues of a real supersymmetric tensor,” Journal of Symbolic Computation, vol. 40, no. 6, pp. 1302–1324, 2005.
  36. M. Zhang, G. Ni, and G. Zhang, “Iterative methods for computing u-eigenvalues of non-symmetric complex tensors with application in quantum entanglement,” Computational Optimization and Applications, vol. 75, pp. 779–798, 2020.
  37. C. Chen, A. Surana, A. M. Bloch, and I. Rajapakse, “Controllability of hypergraphs,” IEEE Transactions on Network Science and Engineering, vol. 8, no. 2, pp. 1646–1657, 2021.
  38. J. Xie and L. Qi, “Spectral directed hypergraph theory via tensors,” Linear and Multilinear Algebra, vol. 64, no. 4, pp. 780–794, 2016.
  39. K. S. Narendra and R. Shorten, “Hurwitz stability of metzler matrices,” IEEE Transactions on Automatic Control, vol. 55, no. 6, pp. 1484–1487, 2010.
  40. A. Mohammadi and M. W. Spong, “Chetaev instability framework for kinetostatic compliance-based protein unfolding,” IEEE Control Systems Letters, vol. 6, pp. 2755–2760, 2022.
  41. C. Chen, “Explicit solutions and stability properties of homogeneous polynomial dynamical systems,” IEEE Transactions on Automatic Control, 2022.
  42. M. Ye, B. D. Anderson, and J. Liu, “Convergence and equilibria analysis of a networked bivirus epidemic model,” SIAM Journal on Control and Optimization, vol. 60, no. 2, pp. S323–S346, 2022.
  43. P. Singh and G. Baruah, “Higher order interactions and species coexistence,” Theoretical Ecology, vol. 14, no. 1, pp. 71–83, 2021.
  44. L. Chen, L. Han, T.-Y. Li, and L. Zhou, “Teneig - tensor eigenpairs solver.” [Online]. Available: https://users.math.msu.edu/users/chenlipi/teneig.html
  45. L. Chen, L. Han, and L. Zhou, “Computing tensor eigenvalues via homotopy methods,” SIAM Journal on Matrix Analysis and Applications, vol. 37, no. 1, pp. 290–319, 2016.
  46. J. Alvarez, I. Orlov, and L. Acho, “An invariance principle for discontinuous dynamic systems with application to a coulomb friction oscillator,” J. Dyn. Sys., Meas., Control, vol. 122, no. 4, pp. 687–690, 2000.
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