Using Maximum Entry-Wise Deviation to Test the Goodness-of-Fit for Stochastic Block Models

TL;DR

This paper introduces a new goodness-of-fit test for stochastic block models based on maximum entry deviation, allowing for growing community numbers and extending to degree-corrected models, with proven asymptotic properties and practical validation.

Contribution

The paper proposes a novel test statistic for stochastic block models that accommodates a linearly growing number of communities and extends to degree-corrected models, with theoretical and empirical validation.

Findings

Test statistic converges to Gumbel distribution under null hypothesis.

Method effectively detects community structure and model fit.

Works well in simulations and real data examples.

Abstract

The stochastic block model is widely used for detecting community structures in network data. How to test the goodness-of-fit of the model is one of the fundamental problems and has gained growing interests in recent years. In this article, we propose a novel goodness-of-fit test based on the maximum entry of the centered and re-scaled adjacency matrix for the stochastic block model. One noticeable advantage of the proposed test is that the number of communities can be allowed to grow linearly with the number of nodes ignoring a logarithmic factor. We prove that the null distribution of the test statistic converges in distribution to a Gumbel distribution, and we show that both the number of communities and the membership vector can be tested via the proposed method. Further, we show that the proposed test has asymptotic power guarantee against a class of alternatives. We also demonstrate that the proposed method can be extended to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.