Testing for Global Network Structure Using Small Subgraph Statistics

TL;DR

This paper introduces a simple, statistically sound test for detecting community structure in networks based on small subgraph frequencies, effective even with weak signals.

Contribution

It proposes a novel test for community detection using three-node subgraph counts, applicable to networks and Gaussian data, with near-optimal detection power.

Findings

Test statistic is asymptotically normal under null hypothesis.

High power against degree-corrected stochastic block models.

Effective in real-world social, citation, and stock return networks.

Abstract

We study the problem of testing for community structure in networks using relations between the observed frequencies of small subgraphs. We propose a simple test for the existence of communities based only on the frequencies of three-node subgraphs. The test statistic is shown to be asymptotically normal under a null assumption of no community structure, and to have power approaching one under a composite alternative hypothesis of a degree-corrected stochastic block model. We also derive a version of the test that applies to multivariate Gaussian data. Our approach achieves near-optimal detection rates for the presence of community structure, in regimes where the signal-to-noise is too weak to explicitly estimate the communities themselves, using existing computationally efficient algorithms. We demonstrate how the method can be effective for detecting structure in social networks, citation networks for scientific articles, and correlations of stock returns between companies on the S\&P 500.