Testing network correlation efficiently via counting trees
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We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the co-occurrences of signed trees for a family of non-isomorphic trees. When the two networks are Erdős-Rényi random graphs 𝒢(n,q) that are either independent or correlated with correlation coefficient ρ, our test runs in n^2+o(1) time and succeeds with high probability as n→∞, provided that nmin{q,1-q}≥ n^-o(1) and ρ^2>α≈ 0.338, where α is Otter's constant so that the number of unlabeled trees with K edges grows as (1/α)^K. This significantly improves the prior work in terms of statistical accuracy, running time, and graph sparsity.
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