Power-laws in recurrence networks from dynamical systems

TL;DR

This paper demonstrates that recurrence networks derived from dynamical systems exhibit power-law degree distributions, with exponents linked to invariant densities, revealing new insights into the geometric properties of complex systems.

Contribution

It analytically and empirically shows the emergence of power-law degree distributions in recurrence networks and clarifies their relation to system invariants.

Findings

Recurrence networks display power-law degree distributions.

Exponent $b$ relates to invariant densities.

Different behaviors observed in continuous systems.

Abstract

Recurrence networks are a novel tool of nonlinear time series analysis allowing the characterisation of higher-order geometric properties of complex dynamical systems based on recurrences in phase space, which are a fundamental concept in classical mechanics. In this Letter, we demonstrate that recurrence networks obtained from various deterministic model systems as well as experimental data naturally display power-law degree distributions with scaling exponents $\gamma$ that can be derived exclusively from the systems' invariant densities. For one-dimensional maps, we show analytically that $\gamma$ is not related to the fractal dimension. For continuous systems, we find two distinct types of behaviour: power-laws with an exponent $\gamma$ depending on a suitable notion of local dimension, and such with fixed $\gamma=1$.