Homology-Preserving Multi-Scale Graph Skeletonization Using Mapper on Graphs

TL;DR

This paper introduces a novel graph visualization method using a topological data analysis tool called mapper, which creates multi-scale, homology-preserving skeletons of graphs to reduce visual clutter and reveal core structures.

Contribution

The paper develops { og}, a variation of mapper for weighted, undirected graphs, enabling multi-scale skeletonization with a single adjustable parameter and providing an interactive software tool.

Findings

Effective visualization of synthetic and real-world graphs.

Preserves core topological structures during skeletonization.

Enables multi-scale analysis through parameter adjustment.

Abstract

Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph structures, even for moderately sized data of a few hundred nodes, due to visual clutter. We propose to apply the mapper construction -- a popular tool in topological data analysis -- to graph visualization, which provides a strong theoretical basis for summarizing the data while preserving their core structures. We develop a variation of the mapper construction targeting weighted, undirected graphs, called {\mog}, which generates homology-preserving skeletons of graphs. We further show how the adjustment of a single parameter enables multi-scale skeletonization of the input graph. We provide a software tool that enables interactive explorations of such skeletons and demonstrate the effectiveness of our method for synthetic and real-world data.