Settling the Sharp Reconstruction Thresholds of Random Graph Matching

TL;DR

This paper establishes sharp thresholds for the ability to recover vertex correspondences in correlated random graphs, revealing an all-or-nothing phase transition in dense Gaussian models and partial thresholds in sparse Erdős-Rényi graphs.

Contribution

It characterizes the precise thresholds for graph matching in Gaussian and Erdős-Rényi models, including the all-or-nothing phenomenon and exact recovery conditions.

Findings

Sharp threshold for partial matching in dense Gaussian graphs

All vertices can be exactly matched above the threshold in Gaussian models

Determined thresholds for exact recovery in Erdős-Rényi graphs

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.