Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{G}(n, p)$</tex> . For dense graphs with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p=n^{-o(1)}$</tex> , we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of the 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p=n^{-\Theta(1)}$</tex> , 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 [1], [2]. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation in [3] 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.