TL;DR
This paper introduces the persistent path Laplacian (PPL), a novel mathematical tool that captures both topological persistence and homotopic shape evolution in data, extending the capabilities of persistent path homology.
Contribution
The paper proposes PPL, which overcomes PPH's limitation by capturing shape evolution during filtration through harmonic and non-harmonic spectra analysis.
Findings
PPL's harmonic spectra recover PPH's topological persistence.
Non-harmonic spectra reveal homotopic shape evolution.
PPL extends the analysis of directed graphs and networks.
Abstract
Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.
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Taxonomy
TopicsTopological and Geometric Data Analysis