Detection Threshold for Correlated Erdős-Rényi Graphs via Densest Subgraph

The problem of detecting edge correlation between two Erdős-Rényi 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 Erdős-Rényi
$\mathbf {G}(n, p)$
(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 Erdős-Rényi graph.