具有部分正确种子的图匹配
Graph Matching with Partially-Correct Seeds
摘要：
图匹配的目的是发现两个边相关图之间的潜在顶点对应关系，并在不同的领域中找到了许多应用。在本文中，我们研究了一个种子图匹配问题，它假设一组种子，即，预映射的顶点对，是预先给出的。虽然大多数以前的工作需要所有的种子是正确的，我们专注于设置的种子是部分正确的。具体地，考虑两个相关图，其边缘独立于父\ER图$\mathcal{G}(n,p)$进行采样。提供两个图的顶点之间的映射作为种子，其中未知的$β$分数是正确的。我们首先分析了一个简单的算法，匹配顶点的基础上的共同种子的数量在$1$跳邻域，然后进一步提出了一个新的算法，使用种子在$2$跳邻域。我们建立非渐近性能保证完美匹配的$1$跳和$2$跳算法，表明我们的新$2$跳算法需要更少的正确种子比$1$跳算法时，图是稀疏的。此外，通过结合我们的新的性能保证的$1$跳和$2$跳算法，我们达到了最知名的结果（在所需的正确种子的分数）在整个范围内的图稀疏性和显着改善以前的结果在\cite{10.14778/2794367.2794371,lubars2018correcting}时，$p\ge n^{-5/6}$。例如，当$p$是一个常数或$p=n^{-3/4}$，我们表明，只有$Ω(\sqrt{n\log n})$正确的种子足以完美匹配，而以前最有名的结果要求$Ω(n)$和$Ω(n^{3/4}\log n)$正确的种子，分别。数值实验证实了我们的理论研究结果，证明了我们的$2$跳算法的优越性，在各种合成和真实的图形。
英文摘要：
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 $β$ 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 $Ω(\sqrt{n\log n})$ correct seeds suffice for perfect matching, while the previously best-known results demand $Ω(n)$ and $Ω(n^{3/4}\log n)$ 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.
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