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

Abstract

A fundamental problem in network data analysis is to test Erdös–Rényi model Gn,a+b2n versus a bisection stochastic block model Gn,an,bn, 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 (Bickel and Sarkar, 2016; Lei, 2016) and community detection (Montanari and Sen, 2016). Existing work has been focusing on growing-degree regime a,b→∞ (Bickel and Sarkar, 2016; Lei, 2016; Montanari and Sen, 2016; Banerjee and Ma, 2017; Banerjee, 2018; Gao and Lafferty, 2017a,b) 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 (1995).