A goodness-of-fit test for stochastic block models

TL;DR

This paper introduces a new goodness-of-fit test for stochastic block models in network analysis, leveraging singular values and random matrix theory to assess model fit and estimate community numbers.

Contribution

It develops a novel test statistic based on the largest singular value and provides asymptotic null distribution and power analysis, enabling consistent community detection.

Findings

Test has full power against finer alternative structures

Provides asymptotic null distribution using random matrix theory

Enables sequential estimation of the number of communities

Abstract

The stochastic block model is a popular tool for studying community structures in network data. We develop a goodness-of-fit test for the stochastic block model. The test statistic is based on the largest singular value of a residual matrix obtained by subtracting the estimated block mean effect from the adjacency matrix. Asymptotic null distribution is obtained using recent advances in random matrix theory. The test is proved to have full power against alternative models with finer structures. These results naturally lead to a consistent sequential testing estimate of the number of communities.