Persistent Homology of Coarse Grained State Space Networks

TL;DR

This paper introduces a topological data analysis method using persistent homology on coarse-grained state space networks, improving dynamic state detection and noise robustness over existing approaches.

Contribution

It presents a novel application of persistent homology to coarse-grained state space networks, enhancing dynamic state detection and computational efficiency.

Findings

CGSSN captures richer dynamic information

Improved noise robustness in state detection

More computationally efficient than traditional TDA methods

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.