  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.
