Settling the Sharp Reconstruction Thresholds of Random Graph Matching

• URL: http://arxiv.org/abs/2102.00082v1
• Date: Fri, 29 Jan 2021 21:49:50 GMT
• Title: Settling the Sharp Reconstruction Thresholds of Random Graph Matching
• Authors: Yihong Wu and Jiaming Xu and Sophie H. Yu
• Abstract summary: We study the problem of recovering the hidden correspondence between two edge-correlated random graphs. For dense graphs with $p=n-o(1)$, we prove that there exists a sharp threshold. For sparse ErdHos-R'enyi graphs with $p=n-Theta(1)$, we show that the all-or-nothing phenomenon no longer holds.
• Score: 19.54246087326288
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Erd\H{o}s-R\'enyi model where the two graphs are subsampled from a common parent Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$. 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 Erd\H{o}s-R\'enyi graphs with $p=n^{-\Theta(1)}$, 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 Erd\H{o}s-R\'enyi graphs. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation 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.

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