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Recovering a hidden community beyond the Kesten–Stigum threshold in O(|E|log*|V|) time

Part of: Decision theory Multivariate analysis

Published online by Cambridge University Press:  26 July 2018

Bruce Hajek ,
Yihong Wu  and
Jiaming Xu
Bruce Hajek*
Affiliation:
University of Illinois at Urbana-Champaign
Yihong Wu*
Affiliation:
Yale University
Jiaming Xu*
Affiliation:
Purdue University
*
* Postal address: Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email address: b-hajek@illinois.edu
** Postal address: Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA.
*** Postal address: Krannert School of Management, Purdue University, West Lafayette, IN 47907, USA.
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Abstract

Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime n1-o(1) ≤ K ≤ o(n). A critical parameter is the effective signal-to-noise ratio λ = K2(p - q)2 / ((n - K)q), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log*n + O(1) iterations, with the total time complexity O(|E|log*n), where log*n is the iterated logarithm of n. Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ (n / logn) (ρBP + o(1)), where ρBP is a function of p / q.

Keywords

Hidden communitybelief propagationmessage passingspectral algorithmshigh-dimensional statistics

MSC classification

Primary: 62H12: Estimation
Secondary: 62C20: Minimax procedures
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 325 - 352
Copyright
Copyright © Applied Probability Trust 2018 

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