Persistent homology of complex networks for dynamic state detection

Audun Myers, Elizabeth Munch, and Firas A. Khasawneh
Phys. Rev. E 100, 022314 – Published 21 August 2019

Abstract

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

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  • Received 19 April 2019

DOI:https://doi.org/10.1103/PhysRevE.100.022314

©2019 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsInterdisciplinary PhysicsNetworksStatistical Physics & Thermodynamics

Authors & Affiliations

Audun Myers

  • Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan 48824, USA

Elizabeth Munch

  • Department of Computational Mathematics, Science, and Engineering and Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA

Firas A. Khasawneh

  • Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan 48824, USA

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Vol. 100, Iss. 2 — August 2019

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