Variational Bayesian Inference and Complexity Control for Stochastic Block Models

TL;DR

This paper introduces a new model selection criterion called ILvb for stochastic block models, based on non-asymptotic marginal likelihood approximation, and implements it via a variational Bayes EM algorithm to improve small network clustering.

Contribution

It proposes the ILvb criterion for better estimation of the number of components in SBM, addressing limitations of existing asymptotic criteria.

Findings

ILvb outperforms ICL in small networks

The variational Bayes EM algorithm efficiently computes ILvb

Improved accuracy in vertex clustering for small networks

Abstract

It is now widely accepted that knowledge can be acquired from networks by clustering their vertices according to connection profiles. Many methods have been proposed and in this paper we concentrate on the Stochastic Block Model (SBM). The clustering of vertices and the estimation of SBM model parameters have been subject to previous work and numerous inference strategies such as variational Expectation Maximization (EM) and classification EM have been proposed. However, SBM still suffers from a lack of criteria to estimate the number of components in the mixture. To our knowledge, only one model based criterion, ICL, has been derived for SBM in the literature. It relies on an asymptotic approximation of the Integrated Complete-data Likelihood and recent studies have shown that it tends to be too conservative in the case of small networks. To tackle this issue, we propose a new criterion that we call ILvb, based on a non asymptotic approximation of the marginal likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm.