Testing Network Structure Using Relations Between Small Subgraph Probabilities

TL;DR

This paper develops a statistical framework for testing network structure using small subgraph frequencies, demonstrating that local patterns can reveal global properties without explicit community detection.

Contribution

It introduces new test statistics based on subgraph relations, proves their asymptotic behavior, and shows their effectiveness in detecting community structures with weaker signals than traditional methods.

Findings

The test statistics follow a central limit theorem under Erdős-Rényi models.

Power of the test approaches one for stochastic block models with unknown communities.

Detection is possible with weaker signals than required for community detection.

Abstract

We study the problem of testing for structure in networks using relations between the observed frequencies of small subgraphs. We consider the statistics \begin{align*} T_3 & =(\text{edge frequency})^3 - \text{triangle frequency}\\ T_2 & =3(\text{edge frequency})^2(1-\text{edge frequency}) - \text{V-shape frequency} \end{align*} and prove a central limit theorem for $(T_2, T_3)$ under an Erd\H{o}s-R\'{e}nyi null model. We then analyze the power of the associated $\chi^2$ test statistic under a general class of alternative models. In particular, when the alternative is a $k$-community stochastic block model, with $k$ unknown, the power of the test approaches one. Moreover, the signal-to-noise ratio required is strictly weaker than that required for community detection. We also study the relation with other statistics over three-node subgraphs, and analyze the error under two natural algorithms for sampling small subgraphs. Together, our results show how global structural characteristics of networks can be inferred from local subgraph frequencies, without requiring the global community structure to be explicitly estimated.