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：我们考虑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。图匹配问题出现在包括计算机视觉、自然语言处理和生物信息学在内的多个领域，尤其是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。与相关Erdos-Renyi（ER）模型（已发展出多种高效算法，其中部分被证明能在恒定边相关条件下实现精确匹配）相比，目前尚无低阶多项式算法能在恒定相关条件下实现相关SBM的精确匹配。本研究通过将Mao等人（2021）的思想扩展至含社区结构的图，提出了一种基于比较各顶点根分区树的匹配算法。该算法利用顶点在不同社区间的边统计量，将每个顶点的大邻域划分为互不相交的子集。我们的算法是首个以高概率在稠密图中实现两个相关SBM精确匹配的低阶多项式时间复杂度算法。