An Eigengap Ratio Test for Determining the Number of Communities in Network Data

TL;DR

This paper introduces a simple, parameter-free eigengap ratio test for accurately determining the number of communities in various network models, effective for both dense and sparse networks.

Contribution

It proposes a novel eigengap ratio test that does not require model estimation or parameter tuning, applicable across diverse network structures and community counts.

Findings

Test statistic converges to Tracy-Widom distribution under null hypothesis

Method is asymptotically powerful for detecting true number of communities

Simulation and real-world data validate effectiveness across network types

Abstract

To characterize the community structure in network data, researchers have introduced various block-type models, including the stochastic block model, degree-corrected stochastic block model, mixed membership block model, degree-corrected mixed membership block model, and others. A critical step in applying these models effectively is determining the number of communities in the network. However, to our knowledge, existing methods for estimating the number of network communities often require model estimations or are unable to simultaneously account for network sparsity and a divergent number of communities. In this paper, we propose an eigengap-ratio based test that address these challenges. The test is straightforward to compute, requires no parameter tuning, and can be applied to a wide range of block models without the need to estimate network distribution parameters. Furthermore, it is effective for both dense and sparse networks with a divergent number of communities. We show that the proposed test statistic converges to a function of the type-I Tracy-Widom distributions under the null hypothesis, and that the test is asymptotically powerful under alternatives. Simulation studies on both dense and sparse networks demonstrate the efficacy of the proposed method. Three real-world examples are presented to illustrate the usefulness of the proposed test.