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Trajectory Data Analyses for Pedestrian Space-time Activity Study

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Published on: February 25, 2013

Beyond distance: quantifying point cloud dynamics with persistent homology and dynamic optimal transport.

Yixin Wang^1,2, Ting Gao^1,2, Jinqiao Duan^3,4

¹School of Mathematics and Statistics, Huazhong University of Science and Technology , Wuhan, Hubei, People's Republic of China.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|June 18, 2026

Summary

This study introduces a new framework to analyze topological tipping in dynamic systems. It uses multi-scale entropy and hypergraph analysis to reveal localized structural changes during phase transitions.

Keywords:

dynamic topological optimal transport hypergraph entropy medical imaging multiscale tipping persistence entropy

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Area of Science:

Complex Systems Science
Topology
Data Analysis

Background:

Topological Optimal Transport (TpOT) unifies various relational aspects but can obscure localized changes during dynamic transitions.
Analyzing dynamic phase transitions in evolving point clouds requires methods sensitive to transient structural reorganizations.

Purpose of the Study:

To develop a hierarchical framework for analyzing topological tipping in time-evolving point clouds.
To overcome the limitations of global scalar distances in capturing localized dynamic structural changes.

Main Methods:

Extending the topological optimal transport (TpOT) distance with a hierarchical dynamic evaluation framework.
Employing a novel topological and hypergraph reconstruction strategy for interpolating spatial geometry and re-computing topological structures.
Introducing multi-scale indicators: macroscopic metrics (topological distortion, persistence entropy) and mesoscopic dual-perspective hypergraph entropy.
Developing a point-level topological field by propagating cycle-level entropy changes onto individual vertices.

Main Results:

The framework successfully interpolates spatial geometry and re-computes valid topological structures, ensuring physical fidelity.
Macroscopic and mesoscopic indicators effectively capture global shifts and sensitive, asynchronous local rewirings.
The point-level topological field provides a detailed view of topological changes at the individual point level.
Demonstrated utility across diverse dynamical systems, biological aggregation models, and neuroimaging data.

Conclusions:

Combining transport-based alignment with multi-scale entropy diagnostics offers a powerful approach for dynamic topological analysis.
The developed framework enhances the understanding of critical transitions and intelligent control in complex systems.
This method provides a robust tool for identifying and analyzing localized topological changes in evolving data.