Independence testing for inhomogeneous random graphs

TL;DR

This paper investigates the problem of testing for independence between inhomogeneous random graphs, revealing a statistical-computational tradeoff and proposing a polynomial-time test for certain models.

Contribution

It introduces a new framework for independence testing in inhomogeneous Erdős-Rényi graphs, including a polynomial-time test for graphon-based models.

Findings

Detectability of correlations depends on regimes, with some requiring exponential algorithms.

A polynomial-time test is proposed for graphon models, ensuring practical applicability.

The study uncovers a statistical vs. computational tradeoff in independence testing.

Abstract

Testing for independence between graphs is a problem that arises naturally in social network analysis and neuroscience. In this paper, we address independence testing for inhomogeneous Erd\H{o}s-R\'{e}nyi random graphs on the same vertex set. We first formulate a notion of pairwise correlations between the edges of these graphs and derive a necessary condition for their detectability. We next show that the problem can exhibit a statistical vs. computational tradeoff, i.e., there are regimes for which the correlations are statistically detectable but may require algorithms whose running time is exponential in n, the number of vertices. Finally, we consider a special case of correlation testing when the graphs are sampled from a latent space model (graphon) and propose an asymptotically valid and consistent test procedure that also runs in time polynomial in n.