  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.
