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Abstract We develop a topological analysis of robust traffic pace patterns using persistent homology. We develop Rips filtrations, parametrized by pace, for a symmetrization of traffic pace along the (naturally) directed edges in a road network. Our symmetrization is inspired by recent work of Turner (2019Algebr. Geom. Topol.191135–1170). Our goal is to construct barcodes which help identify meaningful pace structures, namely connected components or ‘rings’. We develop a case study of our methods using datasets of Manhattan and Chengdu traffic speeds. In order to cope with the computational complexity of these large datasets, we develop an auxiliary application of the directed Louvain neighborhood-finding algorithm. We implement this as a preprocessing step prior to our main persistent homology analysis in order to coarse-grain small topological structures. We finally compute persistence barcodes on these neighborhoods. The persistence barcodes have a metric structure which allows us to both qualitatively and quantitatively compare traffic networks. As an example of the results, we find robust connected pace structures near Midtown bridges connecting Manhattan to the mainland.
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Homotopy, homology, and persistent homology using closure spaces
Abstract. We develop persistent homology in the setting of filtrations of (Cˇech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable.
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- Award ID(s):
- 1764406
- PAR ID:
- 10540215
- Publisher / Repository:
- Journal of Applied and Computational Topology,
- Date Published:
- Journal Name:
- Journal of Applied and Computational Topology
- ISSN:
- 2367-1726
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s41468-024-00183-8
