# TESTING CORRELATION OF UNLABELED RANDOM GRAPHS

ANNALS OF APPLIED PROBABILITY（2023）

Abstract

We study the problem of detecting the edge correlation between two ran-dom 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 Erd6s- Renyi graphs (with edge probability n-o(1)), we determine the sharp thresh-old at which the optimal testing error probability exhibits a phase transition from zero to one as n & RARR; & INFIN;. For sparse Erd6s-Renyi graphs with edge prob-ability n-n(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 Erd6s-Renyi graphs and a careful enu-meration of subpseudoforests that can be assembled from short orbits of the edge permutation.

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Key words

Hypothesis testing,graph matching,unlabeled graphs,Erd6s-Renyi graphs,condi-tional second moment method,cycle decomposition

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