Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport

A team of researchers has published a novel framework for analyzing the dynamics of complex systems represented as point clouds. The work, titled 'Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport,' addresses a key limitation in the existing Topological Optimal Transport (TpOT) distance. While TpOT is powerful for comparing static structures, its single scalar output can obscure the transient, localized changes that occur during dynamic processes like phase transitions. The new method creates a 'geodesic' path between system states, rigorously recomputing valid topological structures at each step to ensure physical fidelity.

Along this path, the framework introduces a suite of multi-scale diagnostic tools. These include macroscopic metrics like Topological Distortion and Persistence Entropy to track global shifts, and a novel mesoscopic tool called dual-perspective Hypergraph Entropy. This hypergraph approach is particularly sensitive, detecting asynchronous local rewiring events that might be missed otherwise. The researchers validated their framework on diverse systems: physical models like Rayleigh-Van der Pol oscillators, high-dimensional biological swarm models (D'Orsogna model), and real-world longitudinal stroke patient fMRI data. The results demonstrate that combining transport-based alignment with these entropy diagnostics provides a powerful lens for understanding how the fundamental 'shape' of data evolves over time.