Persistent homology of complex networks for dynamic state detection

Area of Science:

• Dynamical Systems and Network Science
• Computational Topology
• Data Analysis

Background:

• Time series data from dynamical systems are often represented as graphs.
• Distinguishing between dynamic states like periodic and chaotic behavior is crucial.
• Existing graph-based methods for time series analysis have limitations in distinguishing dynamic states and robustness to noise.

Purpose of the Study:

• To develop and evaluate an alternative topological data analysis (TDA) approach for time series graph representations.
• To demonstrate the utility of persistent homology for creating compressed, multi-scale graph representations.
• To compare the effectiveness of TDA-based summaries against traditional network characteristics.

Main Methods:

• Utilized persistent homology, a TDA tool, on graph representations of time series.
• Constructed graphs from time series using k-nearest-neighbor embedding and the ordinal partition framework.
• Calculated pairwise distances via shortest paths to define simplicial complex filtrations.
• Extracted geometric and entropy point summaries from persistence diagrams.

Main Results:

• The TDA approach yields compressed, multi-scale graph representations.
• Persistent homology effectively distinguishes between dynamic states such as periodic and chaotic behavior.
• Persistence-based point summaries show clearer distinctions and greater robustness to noise compared to existing graph-based scores, particularly with ordinal graphs.

Conclusions:

• The proposed TDA approach, leveraging persistent homology, offers a powerful alternative for analyzing time series of dynamical systems.
• TDA-based topological summaries provide superior performance in distinguishing dynamic behaviors and handling noisy data.
• Combining TDA with ordinal graph constructions enhances robustness and discriminative power.