Community Correlations and Testing Independence Between Binary Graphs

TL;DR

This paper introduces community correlation measures for binary graphs to test independence, providing a statistically valid, convergent, and computationally efficient method validated through simulations and real-world data examples.

Contribution

It proposes a novel community correlation framework for binary graphs, enabling valid independence testing with proven convergence and asymptotic properties.

Findings

Community correlations measure edge association within vertex communities.

The proposed test accurately detects independence in simulations.

Real-data examples demonstrate practical utility of the correlation measures.

Abstract

Graph data has a unique structure that deviates from standard data assumptions, often necessitating modifications to existing methods or the development of new ones to ensure valid statistical analysis. In this paper, we explore the notion of correlation and dependence between two binary graphs. Given vertex communities, we propose community correlations to measure the edge association, which equals zero if and only if the two graphs are conditionally independent within a specific pair of communities. The set of community correlations naturally leads to the maximum community correlation, indicating conditional independence on all possible pairs of communities, and to the overall graph correlation, which equals zero if and only if the two binary graphs are unconditionally independent. We then compute the sample community correlations via graph encoder embedding, proving they converge to their respective population versions, and derive the asymptotic null distribution to enable a fast, valid, and consistent test for conditional or unconditional independence between two binary graphs. The theoretical results are validated through comprehensive simulations, and we provide two real-data examples: one using Enron email networks and another using mouse connectome graphs, to demonstrate the utility of the proposed correlation measures.