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# Analytical framework for recurrence-network analysis of time series

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### 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.

### Most cited references15

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### The structure and function of complex networks

(2003)
Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
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### Complex Network from Pseudoperiodic Time Series: Topology versus Dynamics

(2006)
We construct complex networks from pseudoperiodic time series, with each cycle represented by a single node in the network. We investigate the statistical properties of these networks for various time series and find that time series with different dynamics exhibit distinct topological structures. Specifically, noisy periodic signals correspond to random networks, and chaotic time series generate networks that exhibit small world and scale free features. We show that this distinction in topological structure results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Standard measures of structure in complex networks can therefore be applied to distinguish different dynamic regimes in time series. Application to human electrocardiograms shows that such statistical properties are able to differentiate between the sinus rhythm cardiograms of healthy volunteers and those of coronary care patients.
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### Hierarchical Organization Unveiled by Functional Connectivity in Complex Brain Networks

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### Author and article information

###### Journal
10.1103/PhysRevE.85.046105
1203.4701

Mathematical & Computational physics, Nonlinear & Complex systems