Random graph matching at Otter's threshold via counting chandeliers

TL;DR

This paper introduces a polynomial-time graph matching algorithm that leverages counting chandelier trees to successfully match correlated Erdős-Rényi graphs at a constant correlation level, applicable to both sparse and dense graphs.

Contribution

It presents the first efficient algorithm capable of matching correlated Erdős-Rényi graphs at a fixed correlation threshold, extending applicability beyond previous methods.

Findings

Successfully matches almost all vertices with high probability when $nq oty$ and $ ho^2> ext{Otter's constant}$

Achieves exact matching under an information-theoretic condition

Applicable to both sparse and dense graphs with explicit correlation constant

Abstract

We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erd\H{o}s-R\'enyi graphs $\mathcal{G}(n,q)$ whose edges are correlated through a latent vertex correspondence, we show that this algorithm correctly matches all but a vanishing fraction of the vertices with high probability, provided that $nq\to\infty$ and the edge correlation coefficient $\rho$ satisfies $\rho^2>\alpha \approx 0.338$, where $\alpha$ is Otter's tree-counting constant. Moreover, this almost exact matching can be made exact under an extra condition that is information-theoretically necessary. This is the first polynomial-time graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs. In comparison, previous methods either require $\rho=1-o(1)$ or are restricted to sparse graphs. The crux of the algorithm is a carefully curated family of rooted trees called chandeliers, which allows effective extraction of the graph correlation from the counts of the same tree while suppressing the undesirable correlation between those of different trees.