A likelihood-ratio type test for stochastic block models with bounded degrees

TL;DR

This paper introduces a likelihood-ratio test for stochastic block models with bounded degrees, addressing a gap in existing methods focused on growing degrees, and analyzes its theoretical properties and practical performance.

Contribution

It proposes a novel likelihood-ratio based test for bounded-degree stochastic block models, deriving its limit distribution and power, and evaluates its effectiveness with simulations and real data.

Findings

Limit distribution follows power Poisson laws under null and alternative hypotheses.

The test has well-characterized asymptotic power in the bounded-degree regime.

Monte-Carlo methods help reduce computational costs.

Abstract

A fundamental problem in network data analysis is to test Erd\"{o}s-R\'{e}nyi model $\mathcal{G}\left(n,\frac{a+b}{2n}\right)$ versus a bisection stochastic block model $\mathcal{G}\left(n,\frac{a}{n},\frac{b}{n}\right)$, where $a,b>0$ are constants that represent the expected degrees of the graphs and $n$ denotes the number of nodes. This problem serves as the foundation of many other problems such as testing-based methods for determining the number of communities (\cite{BS16,L16}) and community detection (\cite{MS16}). Existing work has been focusing on growing-degree regime $a,b\to\infty$ (\cite{BS16,L16,MS16,BM17,B18,GL17a,GL17b}) while leaving the bounded-degree regime untreated. In this paper, we propose a likelihood-ratio (LR) type procedure based on regularization to test stochastic block models with bounded degrees. We derive the limit distributions as power Poisson laws under both null and alternative hypotheses, based on which the limit power of the test is carefully analyzed. We also examine a Monte-Carlo method that partly resolves the computational cost issue. The proposed procedures are examined by both simulated and real-world data. The proof depends on a contiguity theory developed by Janson \cite{J95}.