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.
Introduction. -In the last decade, the complex network paradigm has proven to be a fruitful tool for the investigation of complex systems in various areas of science, e.g., the internet and world wide web in computer science, food webs, gene expression and neural networks in biology, and citation networks in social science [1]. The intricate interplay between the structure and dynamics of real networks has received considerable attention [2]. Particularly, synchronization arising by the transfer of dynamical information in complex network topologies has been studied intensively [3]. The application of complex network theory to climate science is a young field, where only few studies have been reported recently [4][5][6][7][8]. The vertices of a climate network are identified with the spatial grid points of an underlying global climate data set. Edges are added between pairs of vertices depending on the degree of statistical interdependence between the corresponding pairs of anomaly time series taken from the climate data set. Climate networks enable novel insights into the topology and dynamics of the climate system over many spatial scales ranging from local properties as the number of first neighbors of a vertex v (the degree centrality k v ) to global network measures such as the clustering coefficient or the average path length. The local degree centrality and related measures have been used to identify supernodes (re-(a) E-mail: donges@pik-potsdam.de gions of high degree centrality) and to associate them to known dynamical interrelations in the atmosphere, called teleconnection patterns, most notably the North Atlantic Oscillation (NAO) [4]. On the global scale, climate networks were found to possess "small-world" properties due to long range connections (edges linking geographically very distant vertices), that stabilize the climate system and enhance the energy and information transfer within it [4]. By studying the prevalence of long range connections in El Niño and La Niña climate networks [5] and the time dependence of the number of stable edges [6], it has been shown very recently, that the El Niño-Southern Oscillation (ENSO) has a strong impact on the stability of the climate system.
Until now, researchers have used the linear crosscorrelation function of pairs of anomaly time series to quantify the degree of statistical interdependence between different spatial regions. But the highly nonlinear processes at work in the climate system call for the application of nonlinear methods to obtain more reliable results. Here we also use mutual information [9] to construct climate networks allowing to capture linear and nonlinear relationships between time series [8]. Furthermore we use a measure of vertex centrality, betweenness (BC), that is defined locally but takes into account global topological information. Combining these two techniques, we uncover peculiar wave-like structures in the BC fields of climate networks constructed from monthly averaged reanalysis and atmosphere-ocean coupled general circulation model (AOGCM) surface air temperature (SAT) data. Akin to the homonymous data highways of the internet, these BC structures form the backbone of the SAT network, bundling most of the energy flow between remote regions. Some major features of the backbone appear to be closely related to surface ocean currents pointing to an essential role of the oceanic surface circulation in stabilizing the climate system by promoting the global flow of energy, mainly in the form of heat. Note that these insights are conceptually new and cannot be obtained using classical methods of climatology such as principal component analysis (PCA) or singular spectrum analysis (SSA) of anomaly fields [10], because these are by design local in a network sense and are not suitable to study local flow measures depending on a global network topology. We have performed intensive statistical tests with various types of surrogates to ensure the robustness of our results. The methodology developed in this letter has the potential to be universally applicable to extract the energy, matter or information flow structure in any spatially extended dynamical system from observations taken from the real world, experiments and simulations. Our results are hence of interest for several branches of physics as well as various applications, e.g., fluid dynamics (turbulence), plasma physics, biological physics (population dynamics, neural networks, cell models). As demonstrated by its application to the climate system, our method is particularly relevant for the analysis of systems with highly heterogeneous boundary conditions and forcing fields, that are found frequently in nature.
Data. -We utilize the monthly averaged global SAT field to construct climate networks, that allows to directly capture the complex dynamics on the interface between ocean and atmosphere due to heat exchange and other local processes. SAT therefore enables us to s
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