This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rényi graph G(n, p). For dense graphs with p = n −o(1) , we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erdős-Rényi graphs with p = n −Θ(1) , we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erdős-Rényi graphs [CK16, CK17].The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation in [WXY20] and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.

We study the problem of detecting the edge correlation between two random graphs with n unlabeled nodes. This is formalized as a hypothesis testing problem, where under the null hypothesis, the two graphs are independently generated; under the alternative, the two graphs are edge-correlated under some latent node correspondence, but have the same marginal distributions as the null. For both Gaussian-weighted complete graphs and dense Erdős-Rényi graphs (with edge probability n −o(1) ), we determine the sharp threshold at which the optimal testing error probability exhibits a phase transition from zero to one as n → ∞. For sparse Erdős-Rényi graphs with edge probability n −Ω(1) , we determine the threshold within a constant factor.The proof of the impossibility results is an application of the conditional second-moment method, where we bound the truncated second moment of the likelihood ratio by carefully conditioning on the typical behavior of the intersection graph (consisting of edges in both observed graphs) and taking into account the cycle structure of the induced random permutation on the edges. Notably, in the sparse regime, this is accomplished by leveraging the pseudoforest structure of subcritical Erdős-Rényi graphs and a careful enumeration of subpseudoforests that can be assembled from short orbits of the edge permutation.

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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 n min{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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