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Modularity and the spread of perturbations in complex dynamical systems.

Artemy Kolchinsky1,2, Alexander J Gates1,2, Luis M Rocha1,2,3

  • 1School of Informatics and Computing, Indiana University, Bloomington, Indiana 47408, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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We developed a new method to identify modules in dynamical systems by measuring how perturbations spread. This approach reveals hidden structures and dynamics, offering a powerful tool for analyzing complex systems.

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Area of Science:

  • Complex Systems Analysis
  • Dynamical Systems Theory
  • Network Science

Background:

  • Understanding modularity is crucial for analyzing complex dynamical systems.
  • Existing methods for community detection in networks may not fully capture dynamical properties.
  • Dynamical systems exhibit modular organization that can change with system states and perturbations.

Purpose of the Study:

  • To propose a novel method for decomposing dynamical systems into modules based on perturbation spreading.
  • To define and quantify "perturbation modularity" as a measure of dynamical organization.
  • To demonstrate the method's ability to capture dynamic modularity across various conditions.

Main Methods:

  • Decomposition of dynamical systems using perturbation spreading.
  • Maximizing "perturbation modularity" via partitions of system variables.
  • Analysis of coarse-grained perturbed trajectories and autocovariance.
  • Generalization of Markov stability for network community detection.

Main Results:

  • The method effectively separates fast intramodular from slow intermodular dynamics.
  • It captures variations in modular organization across system states, time scales, and perturbation types.
  • Demonstrated uncovering of hierarchical modularity in coupled logistic maps.
  • Identified self-organized modularity in coupled map lattices, dependent on initial state and parameters.

Conclusions:

  • The proposed method offers a principled alternative to traditional network community detection.
  • It provides a powerful tool for exploring the modular organization of complex dynamical systems.
  • The approach is versatile, applicable to various system states, time scales, and perturbations.