Persistent Homology of Complex Networks for Dynamic State Detection

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.

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Cited by in corpus (22)

• Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology
• giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
• On the Automatic Parameter Selection for Permutation Entropy
• Topological Persistence Machine of Phase Transitions
• Fractal Dimension Estimation with Persistent Homology: A Comparative Study
• A cyclic perspective on transient gust encounters through the lens of persistent homology
• A robust estimator of mutual information for deep learning interpretability
• Temporal Network Analysis Using Zigzag Persistence
• Inferring symbolic dynamics of chaotic flows from persistence
• Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
• Detecting bifurcations in dynamical systems with CROCKER plots
• Dark soliton detection using persistent homology
• Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis
• Persistent Homology of Coarse Grained State Space Networks
• Evaluating the phase dynamics of coupled oscillators via time-variant topological features
• Topological Signal Processing using the Weighted Ordinal Partition Network
• Causal Emergence in Discrete and Continuous Dynamical Systems
• Homology Groups of Embedded Fractional Brownian Motion
• Emotion recognition in talking-face videos using persistent entropy and neural networks
• Dynamic State Analysis of a Driven Magnetic Pendulum using Ordinal Partition Networks and Topological Data Analysis
• Effects of Correlated Noise on the Performance of Persistence Based Dynamic State Detection Methods
• Graph-based Summary Statistics for Revealing the Stochastic Gravitational Wave Background in Pulsar Timing Arrays