Graph Distances and Clustering

TL;DR

This paper introduces a new vertex-to-vertex distance based on shared connectivity for graph clustering, compares it with existing measures, and evaluates their effectiveness in reflecting intra-cluster density using synthetic graphs.

Contribution

It proposes a novel distance measure rooted in shared connectivity and benchmarks its effectiveness against existing measures in capturing intra-cluster density.

Findings

Jaccard and Otsuka-Ochiai distances accurately reflect intra-cluster density

Shared connectivity-based distance differs from shortest-path measures

Synthetic graph experiments validate the proposed measures' effectiveness

Abstract

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.