Persistent Homology of Time Series through Complex Networks

TL;DR

This paper introduces a unified pipeline for classifying univariate time series using complex networks and persistent homology, demonstrating the impact of network construction and metrics on performance.

Contribution

It systematically compares different network constructions and metrics, showing their influence on classification accuracy and robustness in a unified framework.

Findings

No single network construction is best for all signals.

Diffusion distance outperforms shortest-path in graph comparison.

Persistence features are robust to noise, aligning with stability theorems.

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.