Modularity and the spread of perturbations in complex dynamical systems

TL;DR

This paper introduces a method to identify modular structures in complex dynamical systems by analyzing how perturbations spread, capturing hierarchical and state-dependent modularity beyond traditional network-based community detection.

Contribution

The paper presents a novel perturbation-based modularity measure that generalizes existing community detection methods to dynamical systems, revealing hierarchical and state-dependent modular organization.

Findings

Effectively separates fast intramodular and slow intermodular dynamics

Uncovers hierarchical modular organization in coupled logistic maps

Identifies self-organized modularity dependent on initial conditions and parameters

Abstract

We propose a method to decompose dynamical systems based on the idea that modules constrain the spread of perturbations. We find partitions of system variables that maximize 'perturbation modularity', defined as the autocovariance of coarse-grained perturbed trajectories. The measure effectively separates the fast intramodular from the slow intermodular dynamics of perturbation spreading (in this respect, it is a generalization of the 'Markov stability' method of network community detection). Our approach captures variation of modular organization across different system states, time scales, and in response to different kinds of perturbations: aspects of modularity which are all relevant to real-world dynamical systems. It offers a principled alternative to detecting communities in networks of statistical dependencies between system variables (e.g., 'relevance networks' or 'functional networks'). Using coupled logistic maps, we demonstrate that the method uncovers hierarchical modular organization planted in a system's coupling matrix. Additionally, in homogeneously-coupled map lattices, it identifies the presence of self-organized modularity that depends on the initial state, dynamical parameters, and type of perturbations. Our approach offers a powerful tool for exploring the modular organization of complex dynamical systems.