Strong Detection Threshold for Correlated Erd\H{o}s-R\'enyi Graphs with Constant Average Degree

TL;DR

This paper determines the exact threshold for detecting correlation between two Erdős-Rényi graphs with constant average degree, resolving previous gaps and establishing when detection is information-theoretically feasible.

Contribution

It establishes a sharp, exact detection threshold for correlated Erdős-Rényi graphs with constant average degree, resolving prior uncertainties.

Findings

Detection is possible if and only if s > min{1/√λ, √α}

Sharp information-theoretic threshold established

Resolves a gap between previous studies

Abstract

Consider a pair of correlated Erd\H{o}s-R\'enyi graphs $\mathcal G(n,\tfrac{\lambda}{n};s)$ that are subsampled from a common parent Erd\H{o}s-R\'enyi graph with average degree $\lambda$ and subsampling probability $s$. We establish a sharp information-theoretic threshold for the detection problem between this model and two independent Erd\H{o}s-R\'enyi graphs $\mathcal G(n,\tfrac{\lambda}{n})$, showing that strong detection is information-theoretically possible if and only if $s>\min\{ \tfrac{1}{\sqrt{\lambda}}, \sqrt{\alpha} \}$ where $\alpha\approx 0.338$ is the Otter's constant. Our result resolves a constant gap between arXiv:2203.14573 and arXiv:2008.10097.