Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

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 $p = n^{-α+o(1)}$ for $α\in (0, 1]$ 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.

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