[论文解读] Graph Matching with Partially-Correct Seeds

Liren Yu, Jiaming Xu+1 more|arXiv (Cornell University)|Apr 8, 2020

Graph Theory and Algorithms参考文献 55被引用 9

一句话总结

本文提出了一种基于两跳邻域的新型图匹配算法，即使在部分种子不正确的情况下，也能以远少于以往方法的正确种子数实现完美匹配。该文建立了非渐近性能保证，表明当 $p$ 为常数或 $p = n^{-3/4}$ 时，仅需 $ olimits\Omega(\sqrt{n\log n})$ 个正确种子即可实现完美匹配，优于以往方法所需的 $ olimits\Omega(n)$ 或 $ olimits\Omega(n^{3/4}\log n)$ 个种子。

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 \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.

研究动机与目标

• 为解决在现实应用中仅提供部分正确种子的图匹配挑战。
• 改进现有假设所有种子均正确的种子图匹配算法，该假设在实际中往往不现实。
• 开发一种在稀疏图设置下，仅需极少正确种子即可保持高匹配准确率的方法。
• 在部分种子正确的情况下，为完美匹配建立非渐近理论性能保证。

提出的方法

• 提出一种1跳邻域算法，基于种子集合中共同的1跳邻居数量匹配顶点。
• 引入一种新的2跳邻域算法，利用二阶邻域信息以提高匹配准确率。
• 在基于2跳见证计数导出的权重上应用贪心最大权匹配（GMWM）算法，以选择顶点对。
• 利用浓度不等式和概率界推导出正确匹配的高概率保证。
• 应用并集界和条件概率论证，控制所有顶点对上的误差事件。
• 建立理论条件，证明2跳算法以高概率实现完美匹配。

实验结果

研究问题

• RQ1 当仅提供部分正确种子时，能否成功完成图匹配？
• RQ2 在所需正确种子数量方面，2跳邻域匹配相较于1跳匹配的性能如何？
• RQ3 在部分种子正确的情况下，稀疏Erdős-Rényi图中实现完美匹配所需的最少正确种子数是多少？
• RQ4 能否在部分种子正确条件下，为种子图匹配建立理论性能保证？

主要发现

• 在稀疏图中，2跳算法所需的正确种子数远少于1跳算法，尤其当 $p \geq n^{-5/6}$ 时更为显著。
• 当 $p$ 为常数时，仅需 $ olimits\Omega(\sqrt{n\log n})$ 个正确种子即可实现完美匹配，优于以往所需的 $ olimits\Omega(n)$。
• 当 $p = n^{-3/4}$ 时，该方法仅需 $ olimits\Omega(\sqrt{n\log n})$ 个正确种子，而以往工作要求 $ olimits\Omega(n^{3/4}\log n)$ 个。
• 理论分析表明，在所推导条件下，2跳算法以高概率（$1 - n^{-1}$）实现完美匹配。
• 在合成图和真实图上的数值实验验证了2跳算法相较于现有方法的优越性。
• 本文在图稀疏性的全范围内，建立了所需正确种子比例的最佳已知结果。