Abstract
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.
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Notes
- 1.
A formicarium is an enclosure for keeping ants under semi-natural conditions [40].
- 2.
To say that A ⊂R is locally finite means that for any bounded interval I ⊂R, the cardinality of I ∩ A is finite.
- 3.
The name formigram is a combination of the words formicarium and diagram.
- 4.
If θ X is not continuous at c, then at least one of the relations of θ X(c − ε) ≤ θ X(c) ≥ θ X(c + ε) would be strict for small ε > 0. But if c is a continuity point of θ X, then θ X(c − ε) = θ X(c) = θ X(c + ε) for small ε > 0.
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Acknowledgements
We acknowledge funding from these sources: NSF-RI-1422400, NSF AF 1526513, NSF DMS 1723003, NSF CCF 1740761.
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Kim, W., Mémoli, F., Smith, Z. (2020). Analysis of Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_14
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