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

We introduce a framework for analysing topological tipping in time evolutionary point clouds by extending the recently proposed topological optimal transport (TpOT) distance. While TpOT unifies geometrical, homological and higher-order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously re-computes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (topological distortion and persistence entropy) to capture global shifts, and a novel mesoscopic dual-perspective hypergraph entropy (node-perspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations of physical dynamical systems (Rayleigh-van der Pol limit cycles, double-well cluster fusion), high-dimensional biological aggregation (D'Orsogna model) and longitudinal stroke fMRI data demonstrate the utility of combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis. This article is part of the theme issue 'Critical transitions and intelligent control in complex systems'.