Exact matching of random graphs with constant correlation

Abstract

This paper deals with the problem of graph matching or network alignment for Erdős–Rényi graphs, which can be viewed as a noisy average-case version of the graph isomorphism problem. Let G and \(G'\) be G(n, p) Erdős–Rényi graphs marginally, identified with their adjacency matrices. Assume that G and \(G'\) are correlated such that \({\mathbb {E}}[G_{ij} G'_{ij}] = p(1-\alpha )\). For a permutation \(\pi \) representing a latent matching between the vertices of G and \(G'\), denote by \(G^\pi \) the graph obtained from permuting the vertices of G by \(\pi \). Observing \(G^\pi \) and \(G'\), we aim to recover the matching \(\pi \). In this work, we show that for every \(\varepsilon \in (0,1]\), there is \(n_0>0\) depending on \(\varepsilon \) and absolute constants \(\alpha _0, R > 0\) with the following property. Let \(n \ge n_0\), \((1+\varepsilon ) \log n \le np \le n^{\frac{1}{R \log \log n}}\), and \(0< \alpha < \min (\alpha _0,\varepsilon /4)\). There is a polynomial-time algorithm F such that \({\mathbb {P}}\{F(G^\pi ,G')=\pi \}=1-o(1)\). This is the first polynomial-time algorithm that recovers the exact matching between vertices of correlated Erdős–Rényi graphs with constant correlation with high probability. The algorithm is based on comparison of partition trees associated with the graph vertices.