The Annals of Applied Probability

Belief propagation, robust reconstruction and optimal recovery of block models

Elchanan Mossel, Joe Neeman, and Allan Sly

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Abstract

We consider the problem of reconstructing sparse symmetric block models with two blocks and connection probabilities $a/n$ and $b/n$ for inter- and intra-block edge probabilities, respectively. It was recently shown that one can do better than a random guess if and only if $(a-b)^{2}>2(a+b)$. Using a variant of belief propagation, we give a reconstruction algorithm that is optimal in the sense that if $(a-b)^{2}>C(a+b)$ for some constant $C$ then our algorithm maximizes the fraction of the nodes labeled correctly. Ours is the only algorithm proven to achieve the optimal fraction of nodes labeled correctly. Along the way, we prove some results of independent interest regarding robust reconstruction for the Ising model on regular and Poisson trees.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2211-2256.

Dates
Received: September 2014
Revised: April 2015
First available in Project Euclid: 1 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1472745457

Digital Object Identifier
doi:10.1214/15-AAP1145

Mathematical Reviews number (MathSciNet)
MR3543895

Zentralblatt MATH identifier
1350.05154

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 91D30: Social networks

Keywords
Stochastic block model unsupervised learning belief propagation robust reconstruction

Citation

Mossel, Elchanan; Neeman, Joe; Sly, Allan. Belief propagation, robust reconstruction and optimal recovery of block models. Ann. Appl. Probab. 26 (2016), no. 4, 2211--2256. doi:10.1214/15-AAP1145. https://projecteuclid.org/euclid.aoap/1472745457


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References

  • [1] Alon, N. and Kahale, N. (1997). A spectral technique for coloring random $3$-colorable graphs. SIAM J. Comput. 26 1733–1748.
  • [2] Bickel, P. J. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106 21068–21073.
  • [3] Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79 473–482.
  • [4] Blum, A. and Spencer, J. (1995). Coloring random and semi-random $k$-colorable graphs. J. Algorithms 19 204–234.
  • [5] Boppana, R. B. (1987). Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science 280–285. IEEE, Los Angeles, CA.
  • [6] Borgs, C., Chayes, J., Mossel, E. and Roch, S. (2006). The Kesten–Stigum reconstruction bound is tight for roughly symmetric binary channels. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06) 518–530.
  • [7] Bui, T. N., Chaudhuri, S., Leighton, F. T. and Sipser, M. (1987). Graph bisection algorithms with good average case behavior. Combinatorica 7 171–191.
  • [8] Coja-Oghlan, A. (2010). Graph partitioning via adaptive spectral techniques. Combin. Probab. Comput. 19 227–284.
  • [9] Condon, A. and Karp, R. M. (2001). Algorithms for graph partitioning on the planted partition model. Random Structures Algorithms 18 116–140.
  • [10] Decelle, A., Krzakala, F., Moore, C. and Zdeborová, L. (2011). Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physics Review E 84 066106.
  • [11] Dyer, M. E. and Frieze, A. M. (1989). The solution of some random NP-hard problems in polynomial expected time. J. Algorithms 10 451–489.
  • [12] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410–433.
  • [13] Holland, P. W., Laskey, K. B. and Leinhardt, S. (1983). Stochastic blockmodels: First steps. Social Networks 5 109–137.
  • [14] Janson, S. and Mossel, E. (2004). Robust reconstruction on trees is determined by the second eigenvalue. Ann. Probab. 32 2630–2649.
  • [15] Jerrum, M. and Sorkin, G. B. (1998). The Metropolis algorithm for graph bisection. Discrete Appl. Math. 82 155–175.
  • [16] Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton–Watson processes. Ann. Math. Statist. 37 1463–1481.
  • [17] Krzakala, F., Moore, C., Mossel, E., Neeman, J., Sly, A., Zdeborová, L. and Zhang, P. (2013). Spectral redemption in clustering sparse networks. Proc. Natl. Acad. Sci. USA 110 20935–20940.
  • [18] Leskovec, J., Lang, K. J., Dasgupta, A. and Mahoney, M. W. (2008). Statistical properties of community structure in large social and information networks. In Proceeding of the 17th International Conference on World Wide Web 695–704. ACM, New York.
  • [19] Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge. Available at http://pages.iu.edu/~rdlyons/.
  • [20] Massoulié, L. (2014). Community detection thresholds and the weak Ramanujan property. In STOC’14—Proceedings of the 2014 ACM Symposium on Theory of Computing 694–703. ACM, New York.
  • [21] McSherry, F. (2001). Spectral partitioning of random graphs. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001) 529–537. IEEE Computer Soc., Los Alamitos, CA.
  • [22] Montanari, A., Mossel, E. and Sly, A. (2012). The weak limit of Ising models on locally tree-like graphs. Probab. Theory Related Fields 152 31–51.
  • [23] Mossel, E., Neeman, J. and Sly, A. (2014). Belief propagation, robust reconstruction, and optimal recovery of block models (extended abstract), vol. 35. In JMLR Workshop and Conference Proceedings (COLT Proceedings) 1–35. Barcelona, Spain.
  • [24] Mossel, E., Neeman, J. and Sly, A. (2015). A proof of the block model threshold conjecture. Preprint. Available at arXiv:1311.4115.
  • [25] Mossel, E., Neeman, J. and Sly, A. (2015). Reconstruction and estimation in the planted partition model. Probab. Theory Related Fields 162 431–461.
  • [26] Sly, A. (2011). Reconstruction for the Potts model. Ann. Probab. 39 1365–1406.
  • [27] Snijders, T. A. B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14 75–100.
  • [28] Strogatz, S. H. (2001). Exploring complex networks. Nature 410 268–276.