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 \ER 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 in \cite{10.14778/2794367.2794371,lubars2018correcting} 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.
翻译:与图表匹配的图表旨在查找两个边缘- cor- complor 平面图之间的隐性顶点对应, 并在不同字段中找到许多应用。 在本文中, 我们研究一个种子式平面匹配问题, 假设先提供一套种子, 即预制的顶点- pair 。 虽然大多数先前的工作要求所有种子都正确, 我们侧重于种子部分正确的地方。 具体地说, 考虑两个相关图表, 其边缘独立于母体 $- ER 平面图 $_ mathcal{ G} (n, p) 。 两张平面平面平面图的顶点之间的映射以种子形式提供, 其中一个未知的 $=beta 部分是正确的。 我们首先分析一个简单的算法, 以$$- hop 附近普通种子的数量为基础, 然后进一步提出一个新的算法, 将种子用于20美元 的整块区段。 我们建立非盈利性的表现保证, 以1美元和 2美元平面的平面的平面算显示, 当我们的两个平面的平面算结果需要2美元时, 更精确的平面的平面的平局需要2美元。