# Graph Matching with Partially-Correct Seeds

Liren Yu, Jiaming Xu, Xiaojun Lin.

Year: 2021, Volume: 22, Issue: 280, Pages: 1−54

#### Abstract

Graph matching aims to find the latent vertex correspondence between two edge-correlated graphs and has found numerous applications across different fields. In this paper, we study a seeded graph matching problem, which assumes that a set of seeds, i.e., pre-mapped vertex-pairs, is given in advance. While most previous work requires all seeds to be correct, we focus on the setting where the seeds are partially correct. Specifically, consider two correlated graphs whose edges are sampled independently from a parent Erdos-Renyi graph $\mathcal{G}(n,p)$. A mapping between the vertices of the two graphs is provided as seeds, of which an unknown $\beta$ fraction is correct. We first analyze a simple algorithm that matches vertices based on the number of common seeds in the $1$-hop neighborhoods, and then further propose a new algorithm that uses seeds in the $2$-hop neighborhoods. We establish non-asymptotic performance guarantees of perfect matching for both $1$-hop and $2$-hop algorithms, showing that our new $2$-hop algorithm requires substantially fewer correct seeds than the $1$-hop algorithm when graphs are sparse. Moreover, by combining our new performance guarantees for the $1$-hop and $2$-hop algorithms, we attain the best-known results (in terms of the required fraction of correct seeds) across the entire range of graph sparsity and significantly improve the previous results when $p\ge n^{-5/6}$. For instance, when $p$ is a constant or $p=n^{-3/4}$, we show that only $\Omega(\sqrt{n\log n})$ correct seeds suffice for perfect matching, while the previously best-known results demand $\Omega(n)$ and $\Omega(n^{3/4}\log n)$ correct seeds, respectively. Numerical experiments corroborate our theoretical findings, demonstrating the superiority of our $2$-hop algorithm on a variety of synthetic and real graphs.