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.

翻译：我们提出了一种实用的计算框架，用于检测参数依赖拓扑数据中的结构变化。在时间序列数据分析、异常检测以及变控制参数系统监测等众多应用中，持久图描述了每个参数值处拓扑特征的诞生与消亡，但未能充分捕捉这些特征随时间演化的重组过程。为克服这一局限，我们将同调特征表示为普通组合Hodge拉普拉斯算子的零模，并在公共环境链空间中追踪对应的特征空间。这使得我们能够计算曲率和和乐，分别作为演化拓扑结构中局部重组与累积记忆的描述符。曲率突出显示同调特征混合或快速变化的参数区域，而和乐则总结了闭合循环后此类变化的净效应。我们还建立了稳定性估计，表明这些描述符在正则区域上对Hodge拉普拉斯算子的扰动具有鲁棒性。在受控时间相关点云数据的数值实验中，所提方法能够检测跟踪不稳定性、区分具有几乎相同持久图的系统，并捕捉逐点特征匹配无法观测的循环级记忆。这些结果表明，零模输运几何可作为分析动态拓扑数据的有用计算工具。