The graph matching problem emerges naturally in various applications such as web privacy, image processing and computational biology. In this paper, graph matching is considered under a stochastic model, where a pair of randomly generated graphs with pairwise correlated edges are to be matched such that given the labeling of the vertices in the first graph, the labels in the second graph are recovered by leveraging the correlation among their edges. The problem is considered under various settings and graph models. In the first step, the Correlated Erd\"{o}s-R\'enyi (CER) graph model is studied, where all edge pairs whose vertices have similar labels are generated based on identical distributions and independently of other edges. A matching scheme called the \textit{typicality matching scheme} is introduced. The scheme operates by investigating the joint typicality of the adjacency matrices of the two graphs. New results on the typicality of permutations of sequences lead to necessary and sufficient conditions for successful matching based on the parameters of the CER model. In the next step, the results are extended to graphs with community structure generated based on the Stochastic Block Model (SBM). The SBM model is a generalization of the CER model where each vertex in the graph is associated with a community label, which affects its edge statistics. The results are further extended to matching of ensembles of more than two correlated graphs. Lastly, the problem of seeded graph matching is investigated where a subset of the labels in the second graph are known prior to matching. In this scenario, in addition to obtaining necessary and sufficient conditions for successful matching, a polytime matching algorithm is proposed.

翻译：图形匹配问题自然出现在网络隐私、 图像处理和计算生物学等各种应用程序中。 在本文中, 图形匹配自然出现在像网络隐私、 图像处理和计算生物学这样的各种应用中。 在本文件中, 图形匹配被考虑在一个随机生成的图形模型中, 将匹配一对配有双对相关边边缘的图表。 鉴于第一个图表中脊柱的标签, 第二图表中的标签会通过利用两个图表边缘的相近性来恢复。 问题会在不同的设置和图形模型中加以考虑 。 在第一步, 研究Cor相关 Erd\\\\\\ { o}s- R\ eny( CER) 图形模型模型模型模型模型模型模型模型模型模型中, 所有的边端配对都有相似的标签。 在下一个步骤中, 匹配结果会扩展到一个社区的直径直的直线, 其直径直的直径直值在Storial 上, 其直径直到直径直径的直径直径直径直到直径直径直径直径直到直径直到直径直的直径直径直径直径直径方。 在Storal的Sto, 上, 的直到直到直的直到直到直到直的直到直至直至直的直至直至直方的直方的直至直至直至直至直方的直方的直方的直至直至直方的直方的直方的直方的直方根直方, 。