Detection Threshold for Correlated Erdős-Rényi Graphs via Densest Subgraph
Jian Ding, Hang Du
Abstract
The problem of detecting edge correlation between two Erdős-Rényi random graphs on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> 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 Erdős-Rényi <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {G}(n, p)$ </tex-math></inline-formula> (so that their marginal distributions are the same as the null). We establish a sharp information-theoretic threshold when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p = n^{-\alpha +o({1)}}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (0, 1]$ </tex-math></inline-formula> 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 Erdős-Rényi graph.