Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
129 tokens/sec
GPT-4o
28 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Community Detection in the Multi-View Stochastic Block Model (2401.09510v1)

Published 17 Jan 2024 in cs.SI, cs.IT, cs.LG, eess.SP, and math.IT

Abstract: This paper considers the problem of community detection on multiple potentially correlated graphs from an information-theoretical perspective. We first put forth a random graph model, called the multi-view stochastic block model (MVSBM), designed to generate correlated graphs on the same set of nodes (with cardinality $n$). The $n$ nodes are partitioned into two disjoint communities of equal size. The presence or absence of edges in the graphs for each pair of nodes depends on whether the two nodes belong to the same community or not. The objective for the learner is to recover the hidden communities with observed graphs. Our technical contributions are two-fold: (i) We establish an information-theoretic upper bound (Theorem~1) showing that exact recovery of community is achievable when the model parameters of MVSBM exceed a certain threshold. (ii) Conversely, we derive an information-theoretic lower bound (Theorem~2) showing that when the model parameters of MVSBM fall below the aforementioned threshold, then for any estimator, the expected number of misclassified nodes will always be greater than one. Our results for the MVSBM recover several prior results for community detection in the standard SBM as well as in multiple independent SBMs as special cases.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (40)
  1. M. E. Newman, “The structure and function of complex networks,” SIAM review, vol. 45, no. 2, pp. 167–256, 2003.
  2. C. Gao, Z. Yin, Z. Wang, X. Li, and X. Li, “Multilayer network community detection: A novel multi-objective evolutionary algorithm based on consensus prior information,” IEEE Computational Intelligence Magazine, vol. 18, no. 2, pp. 46–59, 2023.
  3. M. Lelarge, L. Massoulié, and J. Xu, “Reconstruction in the labelled stochastic block model,” IEEE Transactions on Network Science and Engineering, vol. 2, no. 4, pp. 152–163, 2015.
  4. K. Ahn, K. Lee, H. Cha, and C. Suh, “Binary rating estimation with graph side information,” Advances in Neural Information Processing Systems (NeurIPS), vol. 31, 2018.
  5. Q. Zhang, V. Y. F. Tan, and C. Suh, “Community detection and matrix completion with social and item similarity graphs,” IEEE Transactions on Signal Processing, vol. 69, pp. 917–931, 2021.
  6. Q. Zhang, G. Suh, C. Suh, and V. Y. F. Tan, “Mc2g: An efficient algorithm for matrix completion with social and item similarity graphs,” IEEE Transactions on Signal Processing, vol. 70, pp. 2681–2697, 2022.
  7. O. Sporns, “Contributions and challenges for network models in cognitive neuroscience,” Nature Neuroscience, vol. 17, no. 5, pp. 652–660, 2014.
  8. M. E. Newman, “Detecting community structure in networks,” The European Physical Journal B, vol. 38, pp. 321–330, 2004.
  9. P. W. Holland, K. B. Laskey, and S. Leinhardt, “Stochastic blockmodels: First steps,” Social Networks, vol. 5, no. 2, pp. 109–137, 1983. [Online]. Available: https://www.sciencedirect.com/science/article/pii/0378873383900217
  10. E. Abbe, A. S. Bandeira, and G. Hall, “Exact recovery in the stochastic block model,” IEEE Transactions on Information Theory, vol. 62, no. 1, pp. 471–487, 2015.
  11. E. Abbe and C. Sandon, “Community detection in general stochastic block models: Fundamental limits and efficient algorithms for recovery,” in IEEE Symposium on Foundations of Computer Science (FOCS), 2015, pp. 670–688.
  12. P. Chin, A. Rao, and V. Vu, “Stochastic block model and community detection in sparse graphs: A spectral algorithm with optimal rate of recovery,” in Conference on Learning Theory (COLT), 2015, pp. 391–423.
  13. C. Gao, Z. Ma, A. Y. Zhang, and H. H. Zhou, “Achieving optimal misclassification proportion in stochastic block models,” The Journal of Machine Learning Research, vol. 18, no. 1, pp. 1980–2024, 2017.
  14. H. Saad and A. Nosratinia, “Community detection with side information: Exact recovery under the stochastic block model,” IEEE Journal of Selected Topics in Signal Processing, vol. 12, no. 5, pp. 944–958, 2018.
  15. E. Abbe, J. Fan, K. Wang, and Y. Zhong, “Entrywise eigenvector analysis of random matrices with low expected rank,” The Annals of Statistics, vol. 48, no. 3, pp. 1452–1474, 2020.
  16. M. Esmaeili, H. M. Saad, and A. Nosratinia, “Semidefinite programming for community detection with side information,” IEEE Transactions on Network Science and Engineering, vol. 8, no. 2, pp. 1957–1973, 2021.
  17. M. Esmaeili and A. Nosratinia, “Community detection with known, unknown, or partially known auxiliary latent variables,” IEEE Transactions on Network Science and Engineering, vol. 10, no. 1, pp. 286–304, 2022.
  18. Q. Zhang and V. Y. F. Tan, “Exact recovery in the general hypergraph stochastic block model,” IEEE Transactions on Information Theory, vol. 69, no. 1, pp. 453–471, 2022.
  19. I. E. Chien, C.-Y. Lin, and I.-H. Wang, “On the minimax misclassification ratio of hypergraph community detection,” IEEE Transactions on Information Theory, vol. 65, no. 12, pp. 8095–8118, 2019.
  20. J. Sima, F. Zhao, and S.-L. Huang, “Exact recovery in the balanced stochastic block model with side information,” in Information Theory Workshop (ITW), 2021, pp. 1–6.
  21. X. Li, M. Chen, and Q. Wang, “Multiview-based group behavior analysis in optical image sequence,” Scientia Sinica Informationis, vol. 48, no. 9, pp. 1227–1241, 2018.
  22. H. Tu, C. Wang, and W. Zeng, “Voxelpose: Towards multi-camera 3d human pose estimation in wild environment,” in European Conference on Computer Vision (ECCV), 2020, pp. 197–212.
  23. E. Mossel, J. Neeman, and A. Sly, “Consistency thresholds for the planted bisection model,” in ACM Symposium on Theory of Computing (STOC), 2015, pp. 69–75.
  24. V. Jog and P.-L. Loh, “Information-theoretic bounds for exact recovery in weighted stochastic block models using the renyi divergence,” arXiv preprint arXiv:1509.06418, 2015.
  25. E. Abbe, “Community detection and stochastic block models: recent developments,” The Journal of Machine Learning Research, vol. 18, no. 1, pp. 6446–6531, 2017.
  26. E. Mossel, J. Neeman, and A. Sly, “Reconstruction and estimation in the planted partition model,” Probability Theory and Related Fields, vol. 162, pp. 431–461, 2015.
  27. M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, “Multilayer networks,” Journal of Complex Networks, vol. 2, no. 3, pp. 203–27an application to a 1, 2014.
  28. Z. Ma and S. Nandy, “Community detection with contextual multilayer networks,” IEEE Transactions on Information Theory, vol. 69, no. 5, pp. 3203–3239, 2023.
  29. S. Chen, S. Liu, and Z. Ma, “Global and individualized community detection in inhomogeneous multilayer networks,” The Annals of Statistics, vol. 50, no. 5, pp. 2664–2693, 2022.
  30. N. Stanley, S. Shai, D. Taylor, and P. J. Mucha, “Clustering network layers with the strata multilayer stochastic block model,” IEEE Transactions on Network Science and Engineering, vol. 3, no. 2, pp. 95–105, 2016.
  31. J. Lei, A. R. Zhang, and Z. Zhu, “Computational and statistical thresholds in multi-layer stochastic block models,” arXiv preprint arXiv:2311.07773, 2023.
  32. A. Chatterjee, S. Nandy, and R. Sadhu, “Clustering network vertices in sparse contextual multilayer networks,” arXiv preprint arXiv:2209.07554, 2022.
  33. T. Yang, Y. Chi, S. Zhu, Y. Gong, and R. Jin, “Detecting communities and their evolutions in dynamic social networks—a bayesian approach,” Machine learning, vol. 82, pp. 157–189, 2011.
  34. M. Gosak, R. Markovič, J. Dolenšek, M. S. Rupnik, M. Marhl, A. Stožer, and M. Perc, “Network science of biological systems at different scales: A review,” Physics of Life Reviews, vol. 24, pp. 118–135, 2018.
  35. C. De Bacco, E. A. Power, D. B. Larremore, and C. Moore, “Community detection, link prediction, and layer interdependence in multilayer networks,” Physical Review E, vol. 95, no. 4, p. 042317, 2017.
  36. M. Z. Rácz and A. Sridhar, “Correlated randomly growing graphs,” The Annals of Applied Probability, vol. 32, no. 2, pp. 1058–1111, 2022.
  37. M. Racz and A. Sridhar, “Correlated stochastic block models: Exact graph matching with applications to recovering communities,” Advances in Neural Information Processing Systems (NeurIPS), vol. 34, pp. 22 259–22 273, 2021.
  38. J. Gaudio, M. Z. Racz, and A. Sridhar, “Exact community recovery in correlated stochastic block models,” in Conference on Learning Theory (COLT), 2022, pp. 2183–2241.
  39. E. Onaran, S. Garg, and E. Erkip, “Optimal de-anonymization in random graphs with community structure,” in Asilomar Conference on Signals, Systems and Computers (ACSSC), 2016, pp. 709–713.
  40. S.-Y. Yun and A. Proutiere, “Optimal cluster recovery in the labeled stochastic block model,” Advances in Neural Information Processing Systems (NeurIPS), vol. 29, 2016.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets