Corrected Bayesian information criterion for stochastic block models

TL;DR

This paper introduces a corrected Bayesian information criterion for stochastic block models, improving community number estimation accuracy and establishing theoretical consistency and Wilks theorem for these models.

Contribution

It proposes a new corrected BIC for community detection, demonstrating its consistency and extending results to degree-corrected stochastic block models.

Findings

The corrected BIC outperforms previous criteria in simulations.

The Wilks theorem is established for stochastic block models.

Consistency conditions are provided for various network types.

Abstract

Estimating the number of communities is one of the fundamental problems in community detection. We re-examine the Bayesian paradigm for stochastic block models and propose a "corrected Bayesian information criterion",to determine the number of communities and show that the proposed estimator is consistent under mild conditions. The proposed criterion improves those used in Wang and Bickel (2016) and Saldana et al. (2017) which tend to underestimate and overestimate the number of communities, respectively. Along the way, we establish the Wilks theorem for stochastic block models. Moreover, we show that, to obtain the consistency of model selection for stochastic block models, we need a so-called "consistency condition". We also provide sufficient conditions for both homogenous networks and non-homogenous networks. The results are further extended to degree corrected stochastic block models. Numerical studies demonstrate our theoretical results.