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

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.