Power-laws in recurrence networks from dynamical systems

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 $γ$ that can be derived exclusively from the systems' invariant densities. For one-dimensional maps, we show analytically that $γ$ is not related to the fractal dimension. For continuous systems, we find two distinct types of behaviour: power-laws with an exponent $γ$ depending on a suitable notion of local dimension, and such with fixed $γ=1$.

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