Topological and Geometric Analysis of Time-Series Complexity Dynamics via Discrete Hodge Decomposition

Traditional time-series analysis methods, such as Fourier and wavelet transforms, excel at identifying frequency components and their temporal localization. While powerful for spectral analysis, these methods do not explicitly capture the global geometric structure of state transitions or the emergence of cyclic (non-conservative) dynamics within the signal. In this paper, we propose a novel geometric framework that encodes the local complexity dynamics of a time series as a simplicial complex. Using a sliding Hann window, we map the signal into a sequence of local power spectral density (PSD) distributions. We construct a Vietoris-Rips complex using the Wasserstein distance to preserve the physical metric of frequency shifts, and define a directed edge flow based on the asymmetry of Kullback-Leibler (KL) divergence. Applying discrete Hodge decomposition to this flow separates the dynamics into gradient, curl, and harmonic components.Baseline experiments with synthetic signals demonstrate that our method robustly discriminates commensurable signals (gradient-dominant), incommensurable quasi-periodic signals (emergence of curl flow), and stochastic noise (curl-dominant decomposition). An exploratory application to empirical photoplethysmography (PPG) data demonstrates the framework's capability to characterize real-world biological fluctuations, showing that PPG trajectory patterns are structurally similar to those of incommensurable quasi-periodic signals. In a pilot study with 53 PPG recordings, the harmonic component showed a statistically significant correlation with heart rate that is not explained by standard heart rate variability (HRV) features, suggesting the framework extracts genuinely novel information from physiological signals. This framework offers a potential new mathematical lens for quantifying and classifying the hidden topological structures of time-series data, laying a foundation for future empirical applications and explorations across diverse scientific domains.