# Persistent Homology of Time Series through Complex Networks

arXiv: https://arxiv.org/abs/2605.01624 · HTML: https://mlsanity.com/paper/2605.01624/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 10 of 10)

- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Persistent Homology of Complex Networks for Dynamic State Detection](https://mlsanity.com/paper/1904.07403/) (2019)
- [Time Series Analysis via Network Science: Concepts and Algorithms](https://mlsanity.com/paper/2110.09887/) (2021)
- [Classification of Stochastic Processes with Topological Data Analysis](https://mlsanity.com/paper/2206.03973/) (2022)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Persistent Homology of Coarse Grained State Space Networks](https://mlsanity.com/paper/2206.02530/) (2022)
- [Persistent homology of featured time series data and its applications](https://mlsanity.com/paper/2405.04796/) (2024)
- [Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis](https://mlsanity.com/paper/1905.04329/) (2019)

## Abstract (arXiv)

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.