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.

翻译：本工作致力于对复杂过渡网络进行拓扑分析，以实现动态状态检测。过渡网络由时间序列数据构建，借助图论工具揭示底层动态系统的信息。然而，传统方法难以概括此类图中存在的复杂拓扑结构。本文利用拓扑数据分析中的持续同调方法研究这些网络的结构，并将基于粗粒度状态空间网络与拓扑数据分析的时间序列动态状态检测，与两种最先进方法（结合拓扑数据分析的序数划分网络，以及对信号时滞嵌入进行持续同调的标准应用）进行对比。结果表明，粗粒度状态空间网络能捕捉底层动态系统丰富的状态信息——相较于序数划分网络，动态状态检测精度与噪声鲁棒性均显著提升。同时，由于粗粒度状态空间网络的计算时间与信号长度不呈线性相关，其在对时间序列的时滞嵌入应用拓扑数据分析时具有更高的计算效率。