Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks
This work addresses the problem of matching noisy network data for researchers in network analysis and statistics, providing theoretical foundations and practical measures.
The authors studied the graph matching problem where one graph is an errorfully observed copy of another, showing that the matching solution is a maximum likelihood estimator under a corrupting channel model. They established consistency conditions for this estimator and applied the results to analyze matchability in various random graph families, with experimental validation on simulated and real-world networks.
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.