Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

03/28/2022

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by   Jian Ding, et al.

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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 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∈ (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.