Analytical framework for recurrence-network analysis of time series

TL;DR

This paper develops a rigorous theoretical foundation for recurrence network analysis of time series, introducing continuous measures of attractor geometry and providing new tools for parameter selection and measure estimation.

Contribution

It introduces a formal analytical framework connecting recurrence network measures to geometric properties of attractors, enhancing understanding and application of the method.

Findings

New continuous measures of attractor geometry are defined.

Analytical expressions for measures are derived for various chaotic systems.

Framework improves measure estimation and parameter choice in recurrence networks.

Abstract

Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. {While there are already many successful applications ranging from medicine to paleoclimatology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an $\varepsilon$-recurrence network as a discrete subnetwork of a "continuous" graph with uncountably many vertices and edges corresponding to the system's attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by $\varepsilon$.} In particular, we introduce local measures such as the $\varepsilon$-clustering coefficient, mesoscopic measures such as $\varepsilon$-motif density, path-based measures such as $\varepsilon$-betweennesses, and global measures such as $\varepsilon$-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of $\varepsilon$ via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasi-periodic motions, and for hyperballs and hypercubes, by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments. More generally, the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science.