Gauge Geometry of Hodge Zero-Mode Transport in Parameter-Dependent Topological Data Analysis

Abstract

We propose a practical computational framework for detecting structural changes in parameter-dependent topological data. In many applications, such as time-series data analysis, anomaly detection, and monitoring of systems under changing control parameters, persistence diagrams describe the birth and death of topological features at each parameter value, but they do not fully capture how these features are reorganized over time. To address this limitation, we represent homological features by zero modes of the ordinary combinatorial Hodge Laplacian and track the corresponding feature spaces in a common ambient chain space. This allows us to compute curvature and holonomy as descriptors of local reorganization and accumulated memory in evolving topological structures. Curvature highlights parameter regions where homological features mix or change rapidly, while holonomy summarizes the net effect of such changes after a closed cycle. We also establish stability estimates showing that these descriptors are robust under perturbations of the Hodge Laplacian on regular regions. Numerical experiments on controlled time-dependent point-cloud data show that the proposed method detects tracking instability, distinguishes systems with nearly identical persistence diagrams, and captures cycle-level memory invisible to pointwise feature matching. These results suggest that zero-mode transport geometry can serve as a useful computational tool for analyzing dynamic topological data.