Persistent Homology of Coarse Grained State Space Networks

Abstract

This work is dedicated to the topological analysis of complex transitional networks for dynamic state detection. Transitional networks are formed from time series data and they leverage graph theory tools to reveal information about the underlying dynamic system. However, traditional tools can fail to summarize the complex topology present in such graphs. In this work, we leverage persistent homology from topological data analysis to study the structure of these networks. We contrast dynamic state detection from time series using a coarse-grained state-space network (CGSSN) and topological data analysis (TDA) to two state of the art approaches: ordinal partition networks (OPNs) combined with TDA and the standard application of persistent homology to the time-delay embedding of the signal. We show that the CGSSN captures rich information about the dynamic state of the underlying dynamical system as evidenced by a significant improvement in dynamic state detection and noise robustness in comparison to OPNs. We also show that because the computational time of CGSSN is not linearly dependent on the signal's length, it is more computationally efficient than applying TDA to the time-delay embedding of the time series.

1. Added Max M. Chumley as an author to correctly match the final published journal version. 2. Added remark and note of caution on using this method with discrete maps. 3. Added experimental analysis using periodic and chaotic driven magnetic pendulum data. 4. Included analysis demonstrating that multi-periodic solutions still result in a single loop in the network. 5. Updated the abstract

References in corpus (5)

• From time series to complex networks: the visibility graph
• Complex network approaches to nonlinear time series analysis
• Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems
• Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
• Topological Signal Processing using the Weighted Ordinal Partition Network

Cited by in corpus (2)

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