Efficient random graph matching via degree profiles

TL;DR

This paper introduces an efficient algorithm for random graph matching that recovers vertex correspondence with high probability under certain degree and difference conditions, improving upon previous methods in terms of speed and accuracy.

Contribution

It develops a polynomial-time algorithm based on degree profile distance statistics that outperforms prior approaches in matching Erdős-Rényi graphs.

Findings

Perfect recovery with high probability when degree d = Ω(log^2 n)

Improved error tolerance δ = O(log^{-2/3}(n)) for dense graphs

Algorithm runs in ilde{O}(n d^2 + n^2) time

Abstract

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erd\H{o}s-R\'{e}nyi graphs $G(n,\frac{d}{n})$. This can be viewed as an average-case and noisy version of the graph isomorphism problem. Under this model, the maximum likelihood estimator is equivalent to solving the intractable quadratic assignment problem. This work develops an $\tilde{O}(n d^2+n^2)$-time algorithm which perfectly recovers the true vertex correspondence with high probability, provided that the average degree is at least $d = \Omega(\log^2 n)$ and the two graphs differ by at most $\delta = O( \log^{-2}(n) )$ fraction of edges. For dense graphs and sparse graphs, this can be improved to $\delta = O( \log^{-2/3}(n) )$ and $\delta = O( \log^{-2}(d) )$ respectively, both in polynomial time. The methodology is based on appropriately chosen distance statistics of the degree profiles (empirical distribution of the degrees of neighbors). Before this work, the best known result achieves $\delta=O(1)$ and $n^{o(1)} \leq d \leq n^c$ for some constant $c$ with an $n^{O(\log n)}$-time algorithm \cite{barak2018nearly} and $\delta=\tilde O((d/n)^4)$ and $d = \tilde{\Omega}(n^{4/5})$ with a polynomial-time algorithm \cite{dai2018performance}.