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Related Experiment Video

Updated: Jul 13, 2025

Image-based Lagrangian Particle Tracking in Bed-load Experiments

Image-based Lagrangian Particle Tracking in Bed-load Experiments

Published on: July 20, 2017

PERSISTENT PATH LAPLACIAN.

Rui Wang¹, Guo-Wei Wei^1,2,3

¹Department of Mathematics, Michigan State University, MI 48824, USA.

Foundations of Data Science (Springfield, Mo.)

|October 16, 2023

Summary

This summary is machine-generated.

Persistent path homology (PPH) tracks network topology, but not shape evolution. Persistent path Laplacian (PPL) overcomes this, capturing both topological persistence and homotopic shape changes in data during filtration.

Keywords:

Persistent homology Primary: 62R40 Secondary: 55N31 persistent Laplacian simultaneous geometric and topological analyses spectral data analysis spectral graph theory topological data analysis

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Area of Science:

Mathematics
Data Analysis
Network Theory

Background:

Path homology offers a mathematical model for directed graphs and networks.
Persistent path homology (PPH) extends this with filtration for asymmetric structures.
PPH is limited to topological persistence, not tracking data shape evolution.

Purpose of the Study:

Introduce persistent path Laplacian (PPL) to address PPH limitations.
Enable tracking of homotopic shape evolution alongside topological persistence.
Provide a comprehensive analysis of data filtration in networks.

Main Methods:

Development of the persistent path Laplacian (PPL) framework.
Utilizing harmonic and non-harmonic spectra of the PPL.
Applying filtration to analyze evolving data structures.

Main Results:

PPL's harmonic spectra fully recover the topological persistence of PPH.
PPL's non-harmonic spectra reveal homotopic shape evolution during filtration.
Demonstrated PPL's capability to capture dynamic data changes.

Conclusions:

PPL enhances persistent homology by incorporating shape evolution.
PPL offers a more complete understanding of data filtration in networks.
This new model advances the analysis of complex, evolving data structures.