We study the problem of detecting the edge correlation between two random graphs with 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 ER graphs, we determine the sharp threshold at which the optimal testing error probability exhibits a phase transition from zero to one. For sparse ER graphs, 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. 


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