Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erd\H{o}s--R\'enyi graphs with edge correlation $1-\alpha$, our algorithm recovers the underlying matching exactly with high probability when $\alpha \le 1 / (\log \log n)^C$, where $n$ is the number of vertices in each graph and $C$ denotes a positive universal constant. This improves the condition $\alpha \le 1 / (\log n)^C$ achieved in previous work.
翻译:图形匹配, 也称为网络对齐, 指在两个指定图形的顶端组之间找到双向径角, 以便最大限度地对齐它们的边缘。 这个基本的计算问题经常出现在计算机视觉和生物学等多个领域。 最近, 在概率模型下对齐图形的高效算法进行了大量研究。 在这项工作中, 我们提出一个新的图表匹配算法 : 我们的算法使用多阶段程序将每个顶端与一个签名矢量连接在一起, 然后匹配两个图形中的一对顶端, 如果它们的签名矢量彼此接近的话。 我们显示, 对于两个具有边缘相关性的 Erd\ H{ { o}s- R\ enyi 图形, 1\\\ alpha$, 我们的算法在$\ alpha\le 1 / (log\log n) $ ($n) 和美元是每个图形中的顶点数, $C 表示一个积极的通用常数。 这改善了在前一个工作中实现的 $\\\\\\\\\ n\\\\\ n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\