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 the perturbation of features that occur in time-varying graphs. Specifically, we slice the graph data into temporal snapshots, transform the snapshots 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 graph . To validate our approach, we examine the stability of these metric spaces by performing a comparative analysis of topological and metric space features under perturbations. Finally, 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  .
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 .
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the perturbation of features that occur
structural changes
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slice the graph data into temporal snapshots, transform the snapshots into
transform each instance of the time-varying graph into
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,
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graph
data
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examine the stability of these metric spaces by performing a comparative analysis of topological and metric space features under perturbations. Finally, we
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. We also examine whether persistence-based similarity measure as a graph metric satisfies a set of well-established, desirable properties for graph metrics