Improved Achievability and Converse Bounds for Erd\H{o}s-R\'enyi Graph Matching

TL;DR

This paper advances the theoretical understanding of graph matching by improving bounds on the conditions needed for perfect vertex correspondence recovery in correlated Erdős-Rényi graphs, highlighting the impact of correlation levels.

Contribution

It improves existing achievability bounds and introduces a new converse bound for exact graph matching, clarifying the threshold dependence on correlation and graph sparsity.

Findings

Improved the achievability bound for graph matching.

Established a new converse bound for impossibility conditions.

Identified a factor of two gap in bounds for sparse, highly correlated graphs.

Abstract

We consider the problem of perfectly recovering the vertex correspondence between two correlated Erd\H{o}s-R\'enyi (ER) graphs. For a pair of correlated graphs on the same vertex set, the correspondence between the vertices can be obscured by randomly permuting the vertex labels of one of the graphs. In some cases, the structural information in the graphs allow this correspondence to be recovered. We investigate the information-theoretic threshold for exact recovery, i.e. the conditions under which the entire vertex correspondence can be correctly recovered given unbounded computational resources. Pedarsani and Grossglauser provided an achievability result of this type. Their result establishes the scaling dependence of the threshold on the number of vertices. We improve on their achievability bound. We also provide a converse bound, establishing conditions under which exact recovery is impossible. Together, these establish the scaling dependence of the threshold on the level of correlation between the two graphs. The converse and achievability bounds differ by a factor of two for sparse, significantly correlated graphs.