Information-Theoretic Thresholds for the Alignments of Partially Correlated Graphs

This paper studies the problem of recovering hidden vertex correspondences between two correlated random graphs. We introduce the partially correlated Erdős-Rényi model and the partially correlated Gaussian Wigner model, where a pair of induced subgraphs is correlated. We investigate the information-theoretic thresholds for recovering these latent correlated subgraphs and their hidden vertex correspondences. For the partially correlated Erdős-Rényi model, we establish the optimal rate for partial recovery: above this threshold, a positive fraction of vertices can be correctly matched, while below it, matching any positive fraction is impossible. We also determine the optimal rate for exact recovery. In the partially correlated Gaussian Wigner model, the optimal rates for partial and exact recovery coincide. To prove the achievability results, we introduce correlated functional digraphs to partition the edges and bound error probabilities using lower-order cumulant generating functions. Our impossibility results rely on a generalized Fano’s inequality and the recovery thresholds for correlated Erdős-Rényi graphs.