Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology
Mustafa Hajij, Bei Wang, Carlos Scheidegger, Paul Rosen
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, highdimensional 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 a metric space, 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 datasets 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 a persistence-based similarity measure satisfies a set of well-established, desirable properties for graph metrics.
