Non-convex exact community recovery in stochastic block model

Abstract

Community detection in graphs that are generated according to stochastic block models (SBMs) has received much attention lately. In this paper, we focus on the binary symmetric SBM-in which a graph of n vertices is randomly generated by first partitioning the vertices into two equal-sized communities and then connecting each pair of vertices with probability that depends on their community memberships-and study the associated exact community recovery problem. Although the maximum-likelihood formulation of the problem is non-convex and discrete, we propose to tackle it using a popular iterative method called projected power iterations. To ensure fast convergence of the method, we initialize it using a point that is generated by another iterative method called orthogonal iterations, which is a classic method for computing invariant subspaces of a symmetric matrix. We show that in the logarithmic sparsity regime of the problem, with high probability the proposed two-stage method can exactly recover the two communities down to the information-theoretic limit in O(n log 2 n/ log log n) time, which is competitive with a host of existing state-of-the-art methods that have the same recovery performance. We also conduct numerical experiments on both synthetic

Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method

“…In particular, when K = 2, the projection operators in these two works both admit a closed-form solution, which can be done via partial sorting. Moreover, our method can be applied to do community detection in the setting of multiple communities, i.e., K ≥ 2, while that in Wang et al (2020) only works when K = 2. Besides, the method in Wang et al (2020) requires a spectral initialization to satisfy a condition of almost exact recovery.…”

“…Moreover, our method can be applied to do community detection in the setting of multiple communities, i.e., K ≥ 2, while that in Wang et al (2020) only works when K = 2. Besides, the method in Wang et al (2020) requires a spectral initialization to satisfy a condition of almost exact recovery. By contrast, any point satisfying the partial recovery condition in ( 7), including some spectral initializations, is a qualified initialization for Algorithm 1.…”

“…Moreover, it recovers the communities exactly with high probability all the way down to information-theoretic threshold in O(n 1+1/ log log n ) time in the considered SBM. Besides, Gao et al (2017) proposed a two-stage algorithm that Wang et al (2020) proposed a non-convex approach that involves initializing a generalized power method with a power method for solving a regularized ML formulation of the binary symmetric SBM. Their method runs in nearlylinear time and is among the most efficient in the literature that achieves exact recovery at the information-theoretic limit.…”

“…Recently, in independent lines of research, many nonconvex formulations that arise in a variety of applications have been shown to be solvable, in the sense of averagecase performance, by simple and scalable iterative methods. This includes phase retrieval (Bendory et al, 2017;Chen et al, 2019), group synchronization (Ling, 2020;Liu et al, 2017a;2020;Zhong & Boumal, 2018), low-rank matrix recovery (Chi et al, 2019), and two-block community detection (Wang et al, 2020). It then naturally motivates the question of whether one can apply a similar simple and scalable method to the discrete optimization problem (MLE).…”

Differentially Private Community Detection for Stochastic Block Models

“…In particular, in the dense regime (the focus of this paper), with p = a log(n)/n and q = b log(n)/n, for some constants a > b > 0, it is known that exact recovery is possible if and only if √ a− √ b > √ r (see [1] for a comprehensive survey). Efficient algorithms for recovering communities have been developed using spectral methods and semi-definite programming (SDP) [8,30,2,29,17,20,3,37].…”