Consistent Bayesian Community Detection

TL;DR

This paper investigates the statistical performance of Bayesian Stochastic Block Models with a focus on community detection when the number of communities is unknown, demonstrating posterior consistency and developing a reversible-jump MCMC algorithm.

Contribution

It introduces a Bayesian SBM with a diagonally dominant connectivity matrix, proving posterior consistency and proposing a novel reversible-jump MCMC method for community detection.

Findings

Posterior concentrates on true community count and memberships asymptotically.

The proposed method performs competitively in finite samples across various scenarios.

The approach effectively handles unknown community numbers with theoretical guarantees.

Abstract

Stochastic Block Models (SBMs) are a fundamental tool for community detection in network analysis. But little theoretical work exists on the statistical performance of Bayesian SBMs, especially when the community count is unknown. This paper studies a special class of SBMs whose community-wise connectivity probability matrix is diagonally dominant, i.e., members of the same community are more likely to connect with one another than with members from other communities. The diagonal dominance constraint is embedded within an otherwise weak prior, and, under mild regularity conditions, the resulting posterior distribution is shown to concentrate on the true community count and membership allocation as the network size grows to infinity. A reversible-jump Markov Chain Monte Carlo posterior computation strategy is developed by adapting the allocation sampler of Mcdaid et al (2013). Finite sample properties are examined via simulation studies in which the proposed method offers competitive estimation accuracy relative to existing methods under a variety of challenging scenarios.