Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks

TL;DR

This paper models inexact graph matching as a maximum likelihood estimation problem under a new corrupting channel model, providing theoretical conditions for consistency and validating on simulated and real networks.

Contribution

It introduces a corrupting channel model linking graph matching to maximum likelihood estimation, with conditions for consistency and practical validation.

Findings

MLE corresponds to the graph matching solution under the model

Conditions for the consistency of the MLE are established

Experimental validation confirms theoretical results on various networks

Abstract

Given a pair of graphs with the same number of vertices, the inexact graph matching problem consists in finding a correspondence between the vertices of these graphs that minimizes the total number of induced edge disagreements. We study this problem from a statistical framework in which one of the graphs is an errorfully observed copy of the other. We introduce a corrupting channel model, and show that in this model framework, the solution to the graph matching problem is a maximum likelihood estimator. Necessary and sufficient conditions for consistency of this MLE are presented, as well as a relaxed notion of consistency in which a negligible fraction of the vertices need not be matched correctly. The results are used to study matchability in several families of random graphs, including edge independent models, random regular graphs and small-world networks. We also use these results to introduce measures of matching feasibility, and experimentally validate the results on simulated and real-world networks.