Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence

We overview recent work on obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), one first constructs a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, it is possible to then obtain a persistence diagram or barcode from this zigzag persistence module. It turns out that these barcodes are stable under perturbations of the input DG under a certain suitable distance between DGs. We also overview how these results are also applicable in the setting of dynamic metric spaces, and describe a computational application to the analysis of flocking behavior.