We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

翻译：我们研究两个相关随机块模型（SBM）之间的图匹配问题，即学习顶点对应关系。图匹配问题出现在计算机视觉、自然语言处理和生物信息学等多个领域，特别地，匹配具有内在社区结构的图对于相关社交网络的去匿名化具有重要意义。相较于已开发出多种高效算法的相关Erdos-Renyi（ER）模型（其中已有若干算法被证明能在恒定边相关条件下实现精确匹配），对于恒定相关性的相关SBM，此前尚未有任何低阶多项式算法能实现精确匹配。本研究通过扩展Mao等人（2021）的思想至带社区结构的图，提出一种基于比较各顶点根部的分区树的高效匹配算法。该算法利用顶点到不同社区的边统计量，将每个顶点的大范围邻域划分为互不相交的子集。本文提出的算法是首个能在高概率下对稠密图中两个相关SBM实现精确匹配的低阶多项式时间算法。