Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology

TL;DR

This paper introduces a novel persistent homology-based method to detect and visualize structural changes in time-varying graphs, enabling pattern recognition and anomaly detection in complex data.

Contribution

It proposes a new approach transforming graphs into metric spaces and applying persistent homology to quantify and visualize their structural evolution over time.

Findings

Successfully identifies cyclic patterns and deviations in real-world data

Detects one-time events and structural anomalies

Validates persistence-based similarity as a reliable graph metric

Abstract

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into metric spaces, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real world data sets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether persistence-based similarity measure as a graph metric satisfies a set of well-established, desirable properties for graph metrics.