Statistical Limits for Testing Correlation of Hypergraphs

TL;DR

This paper establishes the fundamental limits for testing correlation between two hypergraphs under different probabilistic models, revealing how hypergraph size influences test difficulty.

Contribution

It derives sharp information-theoretic thresholds for hypergraph correlation testing under Gaussian-Wigner and Erdős-Rényi models, extending graph testing theory to hypergraphs.

Findings

Threshold decreases as hyperedge size increases

Testing hypergraph correlation is easier than graph correlation for larger hyperedges

Existence of powerful tests above the threshold

Abstract

In this paper, we consider the hypothesis testing of correlation between two $m$-uniform hypergraphs on $n$ unlabelled nodes. Under the null hypothesis, the hypergraphs are independent, while under the alternative hypothesis, the hyperdges have the same marginal distributions as in the null hypothesis but are correlated after some unknown node permutation. We focus on two scenarios: the hypergraphs are generated from the Gaussian-Wigner model and the dense Erd\"{o}s-R\'{e}nyi model. We derive the sharp information-theoretic testing threshold. Above the threshold, there exists a powerful test to distinguish the alternative hypothesis from the null hypothesis. Below the threshold, the alternative hypothesis and the null hypothesis are not distinguishable. The threshold involves $m$ and decreases as $m$ gets larger. This indicates testing correlation of hypergraphs ($m\geq3$) becomes easier than testing correlation of graphs ($m=2$)