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 (MLE). 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. Supplemental files for this article are available online.

摘要

给定一对具有相同顶点数的图，不精确图匹配问题在于找到这些图的顶点之间的对应关系，以最小化诱导边不一致的总数。我们从统计框架研究这个问题，其中一个图是另一个图的错误观察副本。我们引入了一个破坏通道模型，并表明在这个模型框架中，图匹配问题的解决方案是最大似然估计器（MLE）。给出了此 MLE 一致性的必要和充分条件，以及一个宽松的一致性概念，其中顶点的可忽略部分不需要正确匹配。结果用于研究几个随机图族的匹配性，包括边独立模型、随机正则图、和小世界网络。我们还使用这些结果来介绍匹配可行性的措施，并在模拟和现实世界的网络上通过实验验证结果。本文的补充文件可在线获取。