Electronic Journal of Probability

Consistency thresholds for the planted bisection model

Elchanan Mossel, Joe Neeman, and Allan Sly

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The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors.

Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a “replica” stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the case of logarithmic average degree.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 21, 24 pp.

Received: 12 March 2015
Accepted: 27 January 2016
First available in Project Euclid: 11 March 2016

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]

stochastic block model planted partition model consistency threshold phase transition community detection random network

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Mossel, Elchanan; Neeman, Joe; Sly, Allan. Consistency thresholds for the planted bisection model. Electron. J. Probab. 21 (2016), paper no. 21, 24 pp. doi:10.1214/16-EJP4185. https://projecteuclid.org/euclid.ejp/1457706457

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  • [1] E. Abbe, A. S. Bandeira, and G. Hall. Exact recovery in the stochastic block model. arXiv:1405.3267.
  • [2] Arash A. Amini, Aiyou Chen, Peter J. Bickel, and Elizaveta Levina. Pseudo-likelihood methods for community detection in large sparse networks. The Annals of Statistics, 41(4):2097–2122, 08 2013.
  • [3] P.J. Bickel and A. Chen. A nonparametric view of network models and Newman-Girvan and other modularities. Proceedings of the National Academy of Sciences, 106(50):21068–21073, 2009.
  • [4] R.B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In 28th Annual Symposium on Foundations of Computer Science, pages 280–285. IEEE, 1987.
  • [5] T.N. Bui, S. Chaudhuri, F.T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. Combinatorica, 7(2):171–191, 1987.
  • [6] T. Carson and R. Impagliazzo. Hill-climbing finds random planted bisections. In Twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 903–909. Society for Industrial and Applied Mathematics, 2001.
  • [7] A. Coja-Oghlan. Graph partitioning via adaptive spectral techniques. Combinatorics, Probability and Computing, 19(02):227–284, 2010.
  • [8] A. Condon and R.M. Karp. Algorithms for graph partitioning on the planted partition model. Random Structures and Algorithms, 18(2):116–140, 2001.
  • [9] M.E. Dyer and A.M. Frieze. The solution of some random NP-hard problems in polynomial expected time. Journal of Algorithms, 10(4):451–489, 1989.
  • [10] Paul Erdős and Alfréd Rényi. On the strength of connectedness of a random graph. Acta Mathematica Hungarica, 12(1):261–267, 1961.
  • [11] Theodore E. Harris. A lower bound for the critical probability in a certain percolation process. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 56, pages 13–20. Cambridge Univ. Press, 1960.
  • [12] P.W. Holland, K.B. Laskey, and S. Leinhardt. Stochastic blockmodels: First steps. Social Networks, 5(2):109–137, 1983.
  • [13] M. Jerrum and G.B. Sorkin. The Metropolis algorithm for graph bisection. Discrete Applied Mathematics, 82(1-3):155–175, 1998.
  • [14] R. Karp. Reducibility among combinatorial problems. In R. Miller and J. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum Press, 1972.
  • [15] János Komlós and Endre Szemerédi. Limit distribution for the existence of hamiltonian cycles in a random graph. Discrete Mathematics, 43(1):55–63, 1983.
  • [16] Amit Kumar and Ravindran Kannan. Clustering with spectral norm and the k-means algorithm. In 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2010, pages 299–308. IEEE, 2010.
  • [17] Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 367–384. ACM, 2012.
  • [18] Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Constant factor approximation for balanced cut in the pie model. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 41–49. ACM, 2014.
  • [19] Laurent Massoulié. Community detection thresholds and the weak ramanujan property. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 694–703. ACM, 2014.
  • [20] F. McSherry. Spectral partitioning of random graphs. In 42nd IEEE Symposium on Foundations of Computer Science, pages 529–537. IEEE, 2001.
  • [21] E. Mossel, J. Neeman, and A. Sly. Belief propagation, robust reconstruction, and optimal recovery of block models (extended abstract). JMLR Workshop and Conference Proceedings (COLT proceedings), 35:1–35, 2014. Winner of best paper award at COLT 2014.
  • [22] E. Mossel, J. Neeman, and A. Sly. Stochastic block models and reconstruction. Probability Theory and Related Fields, 2014. (to appear).
  • [23] Elchanan Mossel, Joe Neeman, and Allan Sly. A proof of the block model threshold conjecture. (submitted to Combinatorica), 2014.
  • [24] Raj Rao Nadakuditi and Mark E.J. Newman. Graph spectra and the detectability of community structure in networks. Physical Review Letters, 108(18):188701, 2012.
  • [25] Yoav Seginer. The expected norm of random matrices. Combinatorics, Probability and Computing, 9:149–166, 3 2000.
  • [26] Van H. Vu. Spectral norm of random matrices. Combinatorica, 27(6):721–736, 2007.
  • [27] Se-Young Yun and Alexandre Proutiere. Community detection via random and adaptive sampling. arXiv:1402.3072, 2014.