  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 $\gamma=1$.
