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The Persistent Topology of Dynamic Data

Kim, Woojin
http://rave.ohiolink.edu/etdc/view?acc_num=osu1587503336988272

Abstract Details

2020, Doctor of Philosophy, Ohio State University, Mathematics.
We refine the theoretical foundations of Topological Data Analysis (TDA) for the multiscale analysis of dynamic topology, such as dynamic metric spaces or dynamic networks. Motivations include, but are not limited to, the characterization of flocking or swarming behavior of animals and social networks in the human sphere. In order to quantify the differences of such dynamics, we also generalize the Gromov-Hausdorff distance. We not only examine the resulting metric geometry, but also find practical algorithms for approximating those novel distances. To establish our results, we primarily exploit concepts from algebraic topology, metric geometry, combinatorics, and category theory, blending these with ideas of persistence in TDA. More specifically, the main achievements of this thesis include (a) The development of stable and informative invariants that encode spatiotemporal topological features of dynamic metric spaces and dynamic networks, (b) The establishment of a comparison framework for dynamic metric spaces or dynamic networks by extending the Gromov-Hausdorff distance or the Gromov-Wasserstein distance, (c) Generalization of the erosion distance by Patel for quantifying within polynomial time the difference between dynamic metric spaces or more generally multiparameter persistence modules, and (d) Extension of the notion of persistence diagram for summarizing persistence modules over posets, which often arise from dynamic data, from a standpoint of combinatorics and category theory.
Facundo Mémoli (Advisor)
244 p.
Mathematics
dynamic metric space; dynamic network; dynamic topology; spatiotemporal persistent homology; formigram; topological data analysis; persistence diagram

Recommended Citations

Citations

  • Kim, W. (2020). The Persistent Topology of Dynamic Data [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587503336988272

    APA Style (7th edition)

  • Kim, Woojin. The Persistent Topology of Dynamic Data. 2020. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1587503336988272.

    MLA Style (8th edition)

  • Kim, Woojin. "The Persistent Topology of Dynamic Data." Doctoral dissertation, Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1587503336988272

    Chicago Manual of Style (17th edition)

osu1587503336988272
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This open access ETD is published by The Ohio State University and OhioLINK.