Persistent Homology of Time Series through Complex Networks

arXiv stat.ML / 5/5/2026

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Key Points

The paper introduces a unified pipeline that classifies univariate time series by mapping them into complex networks, converting graphs to dissimilarity matrices, and computing persistent homology via Vietoris–Rips filtrations.
It uses persistence diagrams that are transformed into fixed-length representations (persistence landscapes and topological summary statistics), enabling standardized downstream learning across methods.
The study isolates the effects of design decisions by keeping downstream processing consistent, showing that differences in classification performance come primarily from the chosen network construction and distance metric.
Experiments on 12 UCR benchmarks indicate that no single network construction is universally best, the diffusion distance metric is consistently better than shortest-path alternatives, and topological features remain robust as noise increases.
Overall, the results suggest that persistent-homology-based representations provide graceful degradation under noise, aligning with the known stability properties of persistent homology.

Abstract

We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families (visibility (natural and horizontal visibility graphs), transition, and proximity) and the graph is converted to a dissimilarity matrix from which a Vietoris-Rips filtration yields persistence diagrams. These diagrams are vectorized into fixed-length features through persistence landscapes and topological summary statistics. By standardizing the downstream processing, differences in classification performance are attributable to the network construction and distance metric alone. Experiments on twelve UCR benchmarks show that (i) no single construction dominates: the optimal graph type depends on the signal's discriminative structure; (ii) the graph distance metric is a first-order design choice, with diffusion distance uniformly outperforming shortest-path alternatives; and (iii) persistence-based features degrade gracefully under noise, consistent with the classical stability theorem of persistent homology.