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

Abstract

We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, 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 recomputes 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 on 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.

Figures (15)

• Figure 1: Schematic overview of the hypergraph reconstruction and dynamic distortion framework.Row 1 (Initial TpOT): The optimal spatial coupling $\pi^v_\star$ is computed between the source $X_0$ and target $X_1$ point clouds via solving optimal TpOT problem. Row 2 (Geometric Interpolation): Metric interpolation and Multidimensional Scaling (MDS) generate intermediate spatial configurations $X_t$ along the geodesic $t \in [0,1]$. Row 3 (Topological Structure Reconstruction): Persistent homology is computed based on the interpolated geometric point cloud to extract authentic intermediate features. Row 4 (Hypergraph Reconstruction: Measure topological networks $P_t$ are assembled, where nodes represent regions and colored hulls represent topological hyperedges. Row 5 (Dynamic Distortion as Early Warning Indicators)The dynamic distortions ($\mathcal{L}^t$) are rigorously computed by solving the TpOT distances between the global reference source $P_0$ and each intermediate state $P_t$.
• Figure 2: Ground truth evolution of the stochastic RVP oscillator. (Top) Scatter plots of the state space showing the transition from a limit cycle to a monostable point. Four sparse snapshots are chosen as our training density samples for topological optimal transport task (illustrated in red boxes). (Bottom) The corresponding persistence diagrams tracking the birth and death of the 1-dimensional homological feature.
• Figure 3: Baseline(ground truth) sequential evaluation directly computed between the reference state ($h=-1$) and subsequent empirical snapshots. (a) The topological distortion peaks and flattens exactly as the limit cycle collapses. (b) Both persistence entropy and the proposed symmetric hypergraph entropy exhibit a discontinuous jump at the critical point $h=0$. (c) The jump in symmetric entropy is primarily driven by the hyperedge-perspective component ($\mathrm{HE}_E$).To facilitate a direct visual comparison, both perspective entropies are normalized by their respective theoretical upper bounds.
• Figure 4: Experimental validation via TpOT geodesic interpolation. We subsampled the empirical dataset into merely four equidistant keyframes and reconstructed the continuous topological evolution parameterized by $\tau \in [0,1]$. (a) The MDS-based isometric embedding interpolates the intermediate spatial geometries between the sparse keyframes, approximating the collapse of the limit cycle. (b) The dynamic evaluation along the reconstructed geodesic reproduces the hierarchical macro-meso-micro distortion sequence and the abrupt entropic jumps at the critical transition point, closely aligning with the unobserved ground truth dynamics.
• Figure 5: Phenomenological bifurcation in the double-well potential model. (Top) Scatter plots of the state space showing the transition from a bistable regime(two distinct clusters) to a monostable regime(a single fused cluster). Four sparse snapshots are chosen as our training density samples for topological optimal transport task (illustrated in red boxes). (Bottom) The corresponding persistence diagrams tracking the birth and death of the 1-dimensional homological feature.

Theorems & Definitions (13)

• Definition 1: Vertex-Perspective Entropy
• Definition 2: Hyperedge-Perspective Entropy
• Definition 3: Symmetric Hypergraph Entropy
• Theorem 1: Sensitivity to Abrupt Topological Transitions
• Theorem 2: Dual Sensitivity to Topological Transitions
• Theorem 3: Isomorphism Invariance of Topological Entropy
• Theorem 4: Algebraic Topological Upper Bound
• Corollary 1: Dimension-Specific Topological Bound
• proof : Proof of Property \ref{['thm:maximal']}
• proof : Proof of Theorem \ref{['thm:abrupt_change']}