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Complex networks are usually characterized in terms of their topological, spatial, or information-theoretic properties and combinations of the associated metrics are used to discriminate networks into different classes or categories. However, even with the present variety of characteristics at hand it still remains a subject of current research to appropriately quantify a network's complexity and correspondingly discriminate between different types of complex networks, like infrastructure or social networks, on such a basis. Here, we explore the possibility to classify complex networks by means of a statistical complexity measure that has formerly been successfully applied to distinguish different types of chaotic and stochastic time series. It is composed of a network's averaged per-node entropic measure characterizing the network's information content and the associated Jenson-Shannon divergence as a measure of disequilibrium. We study 29 real world networks and show that networks of the same category tend to cluster in distinct areas of the resulting complexity-entropy plane. We demonstrate that within our framework, connectome networks exhibit among the highest complexity while, e.g, transportation and infrastructure networks display significantly lower values. Furthermore, we demonstrate the utility of our framework by applying it to families of random scale-free and Watts-Strogatz model networks. We then show in a second application that the proposed framework is useful to objectively construct threshold-based networks, such as functional climate networks or recurrence networks, by choosing the threshold such that the statistical network complexity is maximized.

Complex network approaches have been successfully applied for studying transport processes in complex systems ranging from road, railway or airline infrastructure over industrial manufacturing to fluid dynamics. Here, we utilize a generic framework for describing the dynamics of geophysical flows such as ocean currents or atmospheric wind fields in terms of Lagrangian flow networks. In this approach, information on the passive advection of particles is transformed into a Markov chain based on transition probabilities of particles between the volume elements of a given partition of space for a fixed time step. We employ perturbation-theoretic methods to investigate the effects of modifications of transport processes in the underlying flow for three different problem classes: efficient absorption (corresponding to particle trapping or leaking), constant input of particles (with additional source terms modeling, e.g., localized contamination), and shifts of the steady state under probability mass conservation (as arising if the background flow is perturbed itself). Our results demonstrate that in all three cases, changes to the steady state solution can be analytically expressed in terms of the eigensystem of the unperturbed flow and the perturbation itself. These results are potentially relevant for developing more efficient strategies for coping with contaminations of fluid or gaseous media such as ocean and atmosphere by oil spills, radioactive substances, non-reactive chemicals or volcanic aerosols., Comment: 9 pages, 3 figures

El Ni\~no exhibits distinct Eastern Pacific (EP) and Central Pacific (CP) types which are commonly, but not always consistently, distinguished from each other by different signatures in equatorial climate variability. Here, we propose an index based on evolving climate networks to objectively discriminate between both flavors by utilizing a scalar-valued evolving climate network measure that quantifies spatial localization and dispersion in El Ni\~no's associated teleconnections. Our index displays a sharp peak (high localization) during EP events, whereas during CP events (larger dispersion) it remains close to the baseline observed during normal periods. In contrast to previous classification schemes, our approach specifically account for El Ni\~no's global impacts. We confirm recent El Ni\~no classifications for the years 1951 to 2014 and assign types to those cases were former works yielded ambiguous results. Ultimately, we study La Ni\~na episodes and demonstrate that our index provides a similar discrimination into two types.

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve global and local statistics associated with the nodes' embedding in a metric space. Comparing the original network's and the resulting surrogates' global characteristics allows to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links. We apply our framework to various real-world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes' spatial embedding. Depending on the actual performance of the proposed null models, the networks are categorized into different classes. Since many real-world complex networks are in fact spatial networks, the proposed approach is relevant for disentangling underlying complex system structure from spatial embedding of nodes in many fields, ranging from social systems over infrastructure and neurophysiology to climatology.

Studying event time series is a powerful approach for analyzing the dynamics of complex dynamical systems in many fields of science. In this paper, we describe the method of event coincidence analysis to provide a framework for quantifying the strength, directionality and time lag of statistical interrelationships between event series. Event coincidence analysis allows to formulate and test null hypotheses on the origin of the observed interrelationships including tests based on Poisson processes or, more generally, stochastic point processes with a prescribed inter-event time distribution and other higher-order properties. Applying the framework to country-level observational data yields evidence that flood events have acted as triggers of epidemic outbreaks globally since the 1950s. Facing projected future changes in the statistics of climatic extreme events, statistical techniques such as event coincidence analysis will be relevant for investigating the impacts of anthropogenic climate change on human societies and ecosystems worldwide., Comment: 18 pages, 4 figures

We introduce the \texttt{pyunicorn} (Pythonic unified complex network and recurrence analysis toolbox) open source software package for applying and combining modern methods of data analysis and modeling from complex network theory and nonlinear time series analysis. \texttt{pyunicorn} is a fully object-oriented and easily parallelizable package written in the language Python. It allows for the construction of functional networks such as climate networks in climatology or functional brain networks in neuroscience representing the structure of statistical interrelationships in large data sets of time series and, subsequently, investigating this structure using advanced methods of complex network theory such as measures and models for spatial networks, networks of interacting networks, node-weighted statistics or network surrogates. Additionally, \texttt{pyunicorn} provides insights into the nonlinear dynamics of complex systems as recorded in uni- and multivariate time series from a non-traditional perspective by means of recurrence quantification analysis (RQA), recurrence networks, visibility graphs and construction of surrogate time series. The range of possible applications of the library is outlined, drawing on several examples mainly from the field of climatology., Comment: 28 pages, 17 figures

In recent years extensive studies on the Earth's climate system have been carried out by means of advanced complex network statistics. The great majority of these studies, however, have been focusing on investigating correlation structures within single climatic fields directly on or parallel to the Earth's surface. Here, we develop a novel approach of node weighted coupled network measures to study correlations between ocean and atmosphere in the Northern Hemisphere extratropics and construct 18 coupled climate networks, each consisting of two subnetworks. In all cases, one subnetwork represents monthly sea-surface temperature (SST) anomalies, while the other is based on the monthly geopotential height (HGT) of isobaric surfaces at different pressure levels covering the troposphere as well as the lower stratosphere. The weighted cross-degree density proves to be consistent with the leading coupled pattern obtained from maximum covariance analysis. Network measures of higher order allow for a further analysis of the correlation structure between the two fields and consistently indicate that in the Northern Hemisphere extratropics the ocean is correlated with the atmosphere in a hierarchical fashion such that large areas of the ocean surface correlate with multiple statistically dissimilar regions in the atmosphere. Ultimately we show that, this observed hierarchy is linked to large-scale atmospheric variability patterns, such as the Pacific North American pattern, forcing the ocean on monthly time scales.

Complex network theory provides a powerful toolbox for studying the structure of statistical interrelationships between multiple time series in various scientific disciplines. In this work, we apply the recently proposed climate network approach for characterizing the evolving correlation structure of the Earth's climate system based on reanalysis data of surface air temperatures. We provide a detailed study on the temporal variability of several global climate network characteristics. Based on a simple conceptual view on red climate networks (i.e., networks with a comparably low number of edges), we give a thorough interpretation of our evolving climate network characteristics, which allows a functional discrimination between recently recognized different types of El Ni\~no episodes. Our analysis provides deep insights into the Earth's climate system, particularly its global response to strong volcanic eruptions and large-scale impacts of different phases of the El Ni\~no Southern Oscillation (ENSO)., Comment: 20 pages, 12 figures

Eigen techniques such as empirical orthogonal function (EOF) or coupled pattern (CP) / maximum covariance analysis have been frequently used for detecting patterns in multivariate climatological data sets. Recently, statistical methods originating from the theory of complex networks have been employed for the very same purpose of spatio-temporal analysis. This climate network (CN) analysis is usually based on the same set of similarity matrices as is used in classical EOF or CP analysis, e.g., the correlation matrix of a single climatological field or the cross-correlation matrix between two distinct climatological fields. In this study, formal relationships as well as conceptual differences between both eigen and network approaches are derived and illustrated using exemplary global precipitation, evaporation and surface air temperature data sets. These results allow to pinpoint that CN analysis can complement classical eigen techniques and provides additional information on the higher-order structure of statistical interrelationships in climatological data. Hence, CNs are a valuable supplement to the statistical toolbox of the climatologist, particularly for making sense out of very large data sets such as those generated by satellite observations and climate model intercomparison exercises., Comment: 18 pages, 11 figures

We introduce a geometric method for identifying the coupling direction between two dynamical systems based on a bivariate extension of recurrence network analysis. Global characteristics of the resulting inter-system recurrence networks provide a correct discrimination for weakly coupled R\"ossler oscillators not yet displaying generalised synchronisation. Investigating two real-world palaeoclimate time series representing the variability of the Asian monsoon over the last 10,000 years, we observe indications for a considerable influence of the Indian summer monsoon on climate in Eastern China rather than vice versa. The proposed approach can be directly extended to studying $K>2$ coupled subsystems., Comment: 13 pages, 10 figures, 2 tables

Network theory provides a rich toolbox consisting of methods, measures, and models for studying the structure and dynamics of complex systems found in nature, society, or technology. Recently, it has been pointed out that many real-world complex systems are more adequately mapped by networks of interacting or interdependent networks, e.g., a power grid showing interdependency with a communication network. Additionally, in many real-world situations it is reasonable to include node weights into complex network statistics to reflect the varying size or importance of subsystems that are represented by nodes in the network of interest. E.g., nodes can represent vastly different surface area in climate networks, volume in brain networks or economic capacity in trade networks. In this letter, combining both ideas, we derive a novel class of statistical measures for analysing the structure of networks of interacting networks with heterogeneous node weights. Using a prototypical spatial network model, we show that the newly introduced node-weighted interacting network measures indeed provide an improved representation of the underlying system's properties as compared to their unweighted analogues. We apply our method to study the complex network structure of cross-boundary trade between European Union (EU) and non-EU countries finding that it provides important information on trade balance and economic robustness., Comment: 7 pages, 5 figures

Synchronisation between coupled oscillatory systems is a common phenomenon in many natural as well as technical systems. Varying the strength of coupling often leads to qualitative changes in the complex dynamics of the mutually coupled systems including different types of synchronisation such as phase, lag, generalised, or even complete synchronisation. Here, we study the geometric signatures of coupling along with the onset of generalised synchronisation between two coupled chaotic oscillators by mapping the systems' individual as well as joint recurrences in phase space to a complex network. For a paradigmatic continuous-time model system, the transitivity properties of the resulting joint recurrence networks display distinct variations associated with changes in the structural similarity between different parts of the considered trajectories. They therefore provide a useful indicator for the emergence of generalised synchronisation. This paper is dedicated to the 25th anniversary of the introduction of recurrence plots by Eckmann et al. (Europhys. Lett. 4 (1987), 973)., Comment: 7 pages, 3 figures

The absence of time-reversal symmetry is a fundamental property of many nonlinear time series. Here, we propose a new set of statistical tests for time series irreversibility based on standard and horizontal visibility graphs. Specifically, we statistically compare the distributions of time-directed variants of the common complex network measures degree and local clustering coefficient. Our approach does not involve surrogate data and is applicable to relatively short time series. We demonstrate its performance for paradigmatic model systems with known time-reversal properties as well as for picking up signatures of nonlinearity in neuro-physiological data., Comment: 6 pages, 5 figures

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.

Network theory provides various tools for investigating the structural or functional topology of many complex systems found in nature, technology and society. Nevertheless, it has recently been realised that a considerable number of systems of interest should be treated, more appropriately, as interacting networks or networks of networks. Here we introduce a novel graph-theoretical framework for studying the interaction structure between subnetworks embedded within a complex network of networks. This framework allows us to quantify the structural role of single vertices or whole subnetworks with respect to the interaction of a pair of subnetworks on local, mesoscopic and global topological scales. Climate networks have recently been shown to be a powerful tool for the analysis of climatological data. Applying the general framework for studying interacting networks, we introduce coupled climate subnetworks to represent and investigate the topology of statistical relationships between the fields of distinct climatological variables. Using coupled climate subnetworks to investigate the terrestrial atmosphere's three-dimensional geopotential height field uncovers known as well as interesting novel features of the atmosphere's vertical stratification and general circulation. Specifically, the new measure "cross-betweenness" identifies regions which are particularly important for mediating vertical wind field interactions. The promising results obtained by following the coupled climate subnetwork approach present a first step towards an improved understanding of the Earth system and its complex interacting components from a network perspective.

When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or infinite region or set of objects of interest. The selection procedure, e.g., formation of a subset or some kind of discretization or aggregation, typically results in individual nodes of the studied network representing quite differently sized parts of the domain of interest. This heterogeneity may induce substantial bias and artifacts in derived network statistics. To avoid this bias, we propose an axiomatic scheme based on the idea of node splitting invariance to derive consistently weighted variants of various commonly used statistical network measures. The practical relevance and applicability of our approach is demonstrated for a number of example networks from different fields of research, and is shown to be of fundamental importance in particular in the study of spatially embedded functional networks derived from time series as studied in, e.g., neuroscience and climatology., Comment: 21 pages, 13 figures

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches., Comment: To be published in International Journal of Bifurcation and Chaos (2011)

We propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system, relying on the nonlinear mutual information of time series analysis and betweenness centrality of complex network theory. We show, that this approach reveals a rich internal structure in complex climate networks constructed from reanalysis and model surface air temperature data. Our novel method uncovers peculiar wave-like structures of high energy flow, that we relate to global surface ocean currents. This points to a major role of the oceanic surface circulation in coupling and stabilizing the global temperature field in the long term mean (140 years for the model run and 60 years for reanalysis data). We find that these results cannot be obtained using classical linear methods of multivariate data analysis, and have ensured their robustness by intensive significance testing., Comment: 6 pages, 5 figures

Complex network theory provides a powerful framework to statistically investigate the topology of local and non-local statistical interrelationships, i.e. teleconnections, in the climate system. Climate networks constructed from the same global climatological data set using the linear Pearson correlation coefficient or the nonlinear mutual information as a measure of dynamical similarity between regions, are compared systematically on local, mesoscopic and global topological scales. A high degree of similarity is observed on the local and mesoscopic topological scales for surface air temperature fields taken from AOGCM and reanalysis data sets. We find larger differences on the global scale, particularly in the betweenness centrality field. The global scale view on climate networks obtained using mutual information offers promising new perspectives for detecting network structures based on nonlinear physical processes in the climate system., Comment: 24 pages, 10 figures