Efficient Algorithms for Exact Graph Matching on Correlated Stochastic Block Models with Constant Correlation
This addresses the challenge of de-anonymizing correlated social networks and other applications with community structure, providing a novel algorithmic advancement over prior work limited to simpler models.
The paper tackles the problem of exact graph matching on correlated stochastic block models (SBMs) with constant correlation, proposing an efficient algorithm that achieves exact matching with high probability in dense graphs, representing the first low-order polynomial-time solution for this scenario.
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.