Seeded graph matching for correlated Erd\H{o}s-R\'enyi graphs

TL;DR

This paper establishes the theoretical consistency of graph matching in correlated Erdős-Rényi graphs, demonstrating that even with partial observations, accurate latent alignment estimation is achievable using Frank-Wolfe methods.

Contribution

It provides new theoretical guarantees for graph matching in correlated Erdős-Rényi graphs, including partial observation scenarios and connections to human connectomics.

Findings

Graph matching is strongly consistent under modest correlation.

Partial observations of vertex alignments suffice for accurate matching.

Frank-Wolfe algorithm naturally incorporates restricted-focus graph matching.

Abstract

Graph matching is an important problem in machine learning and pattern recognition. Herein, we present theoretical and practical results on the consistency of graph matching for estimating a latent alignment function between the vertex sets of two graphs, as well as subsequent algorithmic implications when the latent alignment is partially observed. In the correlated Erd\H{o}s-R\'enyi graph setting, we prove that graph matching provides a strongly consistent estimate of the latent alignment in the presence of even modest correlation. We then investigate a tractable, restricted-focus version of graph matching, which is only concerned with adjacency involving vertices in a partial observation of the latent alignment; we prove that a logarithmic number of vertices whose alignment is known is sufficient for this restricted-focus version of graph matching to yield a strongly consistent estimate of the latent alignment of the remaining vertices. We show how Frank-Wolfe methodology for approximate graph matching, when there is a partially observed latent alignment, inherently incorporates this restricted focus graph matching. Lastly, we illustrate the relationship between seeded graph matching and restricted-focus graph matching by means of an illuminating example from human connectomics.