With a view on graph clustering, we present a definition of vertex-to-vertex distance which is based on shared connectivity. We argue that vertices sharing more connections are closer to each other than vertices sharing fewer connections. Our thesis is centered on the widely accepted notion that strong clusters are formed by high levels of induced subgraph density, where subgraphs represent clusters. We argue these clusters are formed by grouping vertices deemed to be similar in their connectivity. At the cluster level (induced subgraph level), our thesis translates into low mean intra-cluster distances. Our definition differs from the usual shortest-path geodesic distance. In this article, we compare three distance measures from the literature. Our benchmark is the accuracy of each measure's reflection of intra-cluster density, when aggregated (averaged) at the cluster level. We conduct our tests on synthetic graphs generated using the planted partition model, where clusters and intra-cluster density are known in advance. We examine correlations between mean intra-cluster distances and intra-cluster densities. Our numerical experiments show that Jaccard and Otsuka-Ochiai offer very accurate measures of density, when averaged over vertex pairs within clusters.

中文翻译：

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图距离和聚类

从图聚类的角度来看，我们提出了基于共享连接的顶点到顶点距离的定义。我们认为共享更多连接的顶点比共享更少连接的顶点彼此更接近。我们的论文集中在广泛接受的概念上，即强聚类是由高水平的诱导子图密度形成的，其中子图代表聚类。我们认为这些集群是通过将连接性相似的顶点进行分组而形成的。在集群级别（诱导子图级别），我们的论文转化为低平均集群内距离。我们的定义不同于通常的最短路径测地距离。在本文中，我们比较了文献中的三种距离度量。我们的基准是每个度量反映集群内密度的准确性，在集群级别聚合（平均）时。我们对使用种植分区模型生成的合成图进行测试，其中预先知道集群和集群内密度。我们检查平均簇内距离和簇内密度之间的相关性。我们的数值实验表明，Jaccard 和 Otsuka-Ochiai 提供了非常准确的密度测量，当对集群内的顶点对进行平均时。