# Persistent Homology of Complex Networks for Dynamic State Detection

## TL;DR

This paper introduces a topological data analysis method using persistent homology to analyze graph representations of time series from dynamical systems, effectively distinguishing dynamic states like periodic and chaotic behavior.

## Contribution

It develops a novel TDA approach applying persistent homology to graph-based time series representations, improving state detection robustness and clarity over existing methods.

## Key findings

- Persistence-based summaries better distinguish dynamic states.
- Method is more noise-robust than traditional graph scores.
- Approach works with different graph constructions from time series.

## Abstract

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.