Artemy Kolchinsky1,2,*, Alexander Gates1,2 and Luis M. Rocha1,2,3

1School of Informatics, Indiana University, Bloomington IN, USA
2 Cognitive Science Program, Indiana University, Bloomington IN, USA
3Instituto Gulbenkian de Ciencia, Portugal



Citation: Citation: A. Kolchinsky, A. Gates, L.M. Rocha [2015]. "Modularity and the spread of perturbations in complex dynamical systems". Phys. Rev. E 92, 060801(R).

The full text and pdf re-print are available from the Physical Review E site. Due to mathematical notation and graphics, only the abstract is presented here. The arXiv:1509.04386. pre-print is also available.

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.

Keywords: complex systems, complex networks, modularity, community detection, dynamics, dynamical systems