Testing Dependency of Weighted Random Graphs

TL;DR

This paper investigates the problem of detecting dependency between two weighted random graphs through hypothesis testing, establishing thresholds for detectability and revealing a statistical-computational gap with implications for graph analysis.

Contribution

It formulates the dependency detection as a hypothesis test, derives thresholds for when detection is possible, and identifies a fundamental statistical-computational gap using low-degree polynomial analysis.

Findings

Thresholds for optimal testing are established based on graph size and weight distributions.

Detection is information-theoretically possible or impossible depending on parameters.

A statistical-computational gap is identified, suggesting inherent computational challenges.

Abstract

In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically independent, whereas under the alternative, the edges of one graph are dependent on the edges of a uniformly and randomly vertex-permuted version of the other graph. For general edge-weight distributions, we establish thresholds at which optimal testing becomes information-theoretically possible or impossible, as a function of the total number of nodes in the observed graphs and the generative distributions of the weights. Finally, we identify a statistical-computational gap, and present evidence suggesting that this gap is inherent using the framework of low-degree polynomials.