Detection threshold for correlated Erd\H{o}s-R\'enyi graphs via densest subgraphs

TL;DR

This paper establishes a precise information-theoretic threshold for detecting correlation between two Erdős-Rényi graphs, revealing a novel connection to the densest subgraph problem, and improves upon recent related work.

Contribution

It provides a sharp detection threshold for correlated Erdős-Rényi graphs and introduces a novel link between detection and densest subgraph analysis.

Findings

Established a sharp detection threshold for correlation detection.

Connected the detection problem to densest subgraph analysis.

Improved the constant factor in the detection threshold compared to prior work.

Abstract

The problem of detecting edge correlation between two Erd\H{o}s-R\'enyi random graphs on $n$ unlabeled nodes can be formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are sampled independently; under the alternative, the two graphs are independently sub-sampled from a parent graph which is Erd\H{o}s-R\'enyi $\mathbf{G}(n, p)$ (so that their marginal distributions are the same as the null). We establish a sharp information-theoretic threshold when $p = n^{-\alpha+o(1)}$ for $\alpha\in (0, 1]$ which sharpens a constant factor in a recent work by Wu, Xu and Yu. A key novelty in our work is an interesting connection between the detection problem and the densest subgraph of an Erd\H{o}s-R\'enyi graph.