Uncertainty quantification and testing in a stochastic block model with two unequal communities

TL;DR

This paper establishes posterior convergence and testing methods for community detection in a stochastic block model with two unequal communities, allowing for flexible priors and variable class sizes, including zero and full membership.

Contribution

It provides new theoretical results on posterior convergence and credible set conversion in a flexible two-community stochastic block model with various priors and class size configurations.

Findings

Posterior convergence is achieved under various priors.

Credible sets can be converted into confidence sets.

Symmetric testing with posterior odds is consistent.

Abstract

We show posterior convergence for the community structure in the planted bi-section model, for several interesting priors. Examples include where the label on each vertex is iid Bernoulli distributed, with some parameter $r\in(0,1)$. The parameter $r$ may be fixed, or equipped with a beta distribution. We do not have constraints on the class sizes, which might be as small as zero, or include all vertices, and everything in between. This enables us to test between a uniform (Erd\"os-R\'enyi) random graph with no distinguishable community or the planted bi-section model. The exact bounds for posterior convergence enable us to convert credible sets into confidence sets. Symmetric testing with posterior odds is shown to be consistent.