Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erd\H{o}s-R\'enyi Graphs

TL;DR

This paper demonstrates that a canonical labeling algorithm can efficiently align correlated Erdős-Rényi graphs within a specific parameter region, advancing understanding of graph alignment without seed vertices.

Contribution

It identifies a region where a straightforward canonical labeling algorithm successfully aligns correlated Erdős-Rényi graphs without seeds.

Findings

The algorithm succeeds in a specific parameter region.

Vertex degrees are used for initial matching.

Remaining vertices are aligned via bipartite graph algorithm.

Abstract

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erd\H{o}s-R\'enyi graphs. However, very little is known about the existence of efficient algorithms to achieve graph alignment without seeds. In this work we identify a region in which a straightforward $O(n^{11/5} \log n )$-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in aligning correlated Erd\H{o}s-R\'enyi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance alignment (i.e., sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a alignment algorithm for bipartite graphs.