Non-convex exact community recovery in stochastic block model

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 communitie...

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Bibliographic Details
Published in:Mathematical programming 2022-09, Vol.195 (1-2), p.1-37
Main Authors: Wang, Peng, Zhou, Zirui, So, Anthony Man-Cho
Format: Article
Language:English
Subjects:
Apexes
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Graph theory
Information theory
Iterative methods
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Matrices (mathematics)
Numerical Analysis
Probability theory
Recovery
Stochastic models
Subspaces
Theoretical
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Summary: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 and real data sets to demonstrate the efficacy of our proposed method and complement our theoretical development.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01715-1