# Persistent Homology of Complex Networks for Dynamic State Detection

arXiv: https://arxiv.org/abs/1904.07403 · HTML: https://mlsanity.com/paper/1904.07403/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Apr 2019.

## Built on by (most cited first, 14 of 14)

- [Recurrences reveal shared causal drivers of complex time series](https://mlsanity.com/paper/2301.13516/) (2023)
- [A robust estimator of mutual information for deep learning interpretability](https://mlsanity.com/paper/2211.00024/) (2022)
- [PhoGAD: Graph-based Anomaly Behavior Detection with Persistent Homology Optimization](https://mlsanity.com/paper/2401.10547/) (2024)
- [IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations](https://mlsanity.com/paper/2407.17835/) (2024)
- [Persistent Homology of Coarse Grained State Space Networks](https://mlsanity.com/paper/2206.02530/) (2022)
- [Emotion recognition in talking-face videos using persistent entropy and neural networks](https://mlsanity.com/paper/2110.13571/) (2021)
- [Topological Signal Processing using the Weighted Ordinal Partition Network](https://mlsanity.com/paper/2205.08349/) (2022)
- [Molecule Generation for Target Protein Binding with Hierarchical Consistency Diffusion Model](https://mlsanity.com/paper/2503.00975/) (2025)
- [Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis](https://mlsanity.com/paper/2509.08350/) (2025)
- [Unsupervised Assessment of Landscape Shifts Based on Persistent Entropy and Topological Preservation](https://mlsanity.com/paper/2410.04183/) (2024)
- [Dynamical System Parameter Path Optimization using Persistent Homology](https://mlsanity.com/paper/2505.00782/) (2025)
- [Time Series Analysis of Spiking Neural Systems via Transfer Entropy and Directed Persistent Homology](https://mlsanity.com/paper/2508.19048/) (2025)
- [Persistent Homology of Time Series through Complex Networks](https://mlsanity.com/paper/2605.01624/) (2026)
- [TopTimeNet: Topologically-assisted time-series classification model](https://mlsanity.com/paper/2609.39792/) (2026)

## Abstract (arXiv)

In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram---a two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.