Climate networks around the globe are significantly affected by El Niño

Teleconnection Paths via Climate Network Direct Link Detection

Teleconnections describe remote connections (typically thousands of kilometers) of the climate system. These are of great importance in climate dynamics as they reflect the transportation of energy and climate change on global scales (like the El Niño phenomenon). Yet, the path of influence propagation between such remote regions, and weighting associated with different paths, are only partially known. Here we propose a systematic climate network approach to find and quantify the optimal paths between remotely distant interacting locations. Specifically, we separate the correlations between two grid points into direct and indirect components, where the optimal path is found based on a minimal total cost function of the direct links. We demonstrate our method using near surface air temperature reanalysis data, on identifying cross-latitude teleconnections and their corresponding optimal paths. The proposed method may be used to quantify and improve our understanding regarding the emergence of climate patterns on global scales.

Unified functional network and nonlinear time series analysis for complex systems science: The pyunicorn package

We introduce the 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. 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, 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, 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.

Identifying causal gateways and mediators in complex spatio-temporal systems

Identifying regions important for spreading and mediating perturbations is crucial to assess the susceptibilities of spatio-temporal complex systems such as the Earth's climate to volcanic eruptions, extreme events or geoengineering. Here a data-driven approach is introduced based on a dimension reduction, causal reconstruction, and novel network measures based on causal effect theory that go beyond standard complex network tools by distinguishing direct from indirect pathways. Applied to a data set of atmospheric dynamics, the method identifies several strongly uplifting regions acting as major gateways of perturbations spreading in the atmosphere. Additionally, the method provides a stricter statistical approach to pathways of atmospheric teleconnections, yielding insights into the Pacific-Indian Ocean interaction relevant for monsoonal dynamics. Also for neuroscience or power grids, the novel causal interaction perspective provides a complementary approach to simulations or experiments for understanding the functioning of complex spatio-temporal systems with potential applications in increasing their resilience to shocks or extreme events.

Shape anomaly detection under strong measurement noise: An analytical approach to adaptive thresholding

We suggest an analytical approach to the adaptive thresholding in a shape anomaly detection problem. We find an analytical expression for the distribution of the cosine similarity score between a reference shape and an observational shape hindered by strong measurement noise that depends solely on the noise level and is independent of the particular shape analyzed. The analytical treatment is also confirmed by computer simulations and shows nearly perfect agreement. Using this analytical solution, we suggest an improved shape anomaly detection approach based on adaptive thresholding. We validate the noise robustness of our approach using typical shapes of normal and pathological electrocardiogram cycles hindered by additive white noise. We show explicitly that under high noise levels our approach considerably outperforms the conventional tactic that does not take into account variations in the noise level.

Complex network based techniques to identify extreme events and (sudden) transitions in spatio-temporal systems

We present here two promising techniques for the application of the complex network approach to continuous spatio-temporal systems that have been developed in the last decade and show large potential for future application and development of complex systems analysis. First, we discuss the transforming of a time series from such systems to a complex network. The natural approach is to calculate the recurrence matrix and interpret such as the adjacency matrix of an associated complex network, called recurrence network. Using complex network measures, such as transitivity coefficient, we demonstrate that this approach is very efficient for identifying qualitative transitions in observational data, e.g., when analyzing paleoclimate regime transitions. Second, we demonstrate the use of directed spatial networks constructed from spatio-temporal measurements of such systems that can be derived from the synchronized-in-time occurrence of extreme events in different spatial regions. Although there are many possibilities to investigate such spatial networks, we present here the new measure of network divergence and how it can be used to develop a prediction scheme of extreme rainfall events.

A Network of Networks Perspective on Global Trade

Mutually intertwined supply chains in contemporary economy result in a complex network of trade relationships with a highly non-trivial topology that varies with time. In order to understand the complex interrelationships among different countries and economic sectors, as well as their dynamics, a holistic view on the underlying structural properties of this network is necessary. This study employs multi-regional input-output data to decompose 186 national economies into 26 industry sectors and utilizes the approach of interdependent networks to analyze the substructure of the resulting international trade network for the years 1990-2011. The partition of the network into national economies is observed to be compatible with the notion of communities in the sense of complex network theory. By studying internal versus cross-subgraph contributions to established complex network metrics, new insights into the architecture of global trade are obtained, which allow to identify key elements of global economy. Specifically, financial services and business activities dominate domestic trade whereas electrical and machinery industries dominate foreign trade. In order to further specify each national sector's role individually, (cross-)clustering coefficients and cross-betweenness are obtained for different pairs of subgraphs. The corresponding analysis reveals that specific industrial sectors tend to favor distinct directionality patterns and that the cross-clustering coefficient for geographically close country pairs is remarkably high, indicating that spatial factors are still of paramount importance for the organization of trade patterns in modern economy. Regarding the evolution of the trade network's substructure, globalization is well-expressed by trends of several structural characteristics (e.g., link density and node strength) in the interacting network framework. Extreme events, such as the financial crisis 2008/2009, are manifested as anomalies superimposed to these trends. The marked reorganization of trade patterns, associated with this economic crisis in comparison to "normal" annual fluctuations in the network structure is traced and quantified by a new widely applicable generalization of the Hamming distance to weighted networks.

Most probable paths in temporal weighted networks: An application to ocean transport

We consider paths in weighted and directed temporal networks, introducing tools to compute sets of paths of high probability. We quantify the relative importance of the most probable path between two nodes with respect to the whole set of paths and to a subset of highly probable paths that incorporate most of the connection probability. These concepts are used to provide alternative definitions of betweenness centrality. We apply our formalism to a transport network describing surface flow in the Mediterranean sea. Despite the full transport dynamics is described by a very large number of paths we find that, for realistic time scales, only a very small subset of high probability paths (or even a single most probable one) is enough to characterize global connectivity properties of the network.

Flow networks: a characterization of geophysical fluid transport

We represent transport between different regions of a fluid domain by flow networks, constructed from the discrete representation of the Perron-Frobenius or transfer operator associated to the fluid advection dynamics. The procedure is useful to analyze fluid dynamics in geophysical contexts, as illustrated by the construction of a flow network associated to the surface circulation in the Mediterranean sea. We use network-theory tools to analyze the flow network and gain insights into transport processes. In particular, we quantitatively relate dispersion and mixing characteristics, classically quantified by Lyapunov exponents, to the degree of the network nodes. A family of network entropies is defined from the network adjacency matrix and related to the statistics of stretching in the fluid, in particular, to the Lyapunov exponent field. Finally, we use a network community detection algorithm, Infomap, to partition the Mediterranean network into coherent regions, i.e., areas internally well mixed, but with little fluid interchange between them.

Assessing the direction of climate interactions by means of complex networks and information theoretic tools

An estimate of the net direction of climate interactions in different geographical regions is made by constructing a directed climate network from a regular latitude-longitude grid of nodes, using a directionality index (DI) based on conditional mutual information (CMI). Two datasets of surface air temperature anomalies-one monthly averaged and another daily averaged-are analyzed and compared. The network links are interpreted in terms of known atmospheric tropical and extra-tropical variability patterns. Specific and relevant geographical regions are selected, the net direction of propagation of the atmospheric patterns is analyzed, and the direction of the inferred links is validated by recovering some well-known climate variability structures. These patterns are found to be acting at various time-scales, such as atmospheric waves in the extratropics or longer range events in the tropics. This analysis demonstrates the capability of the DI measure to infer the net direction of climate interactions and may contribute to improve the present understanding of climate phenomena and climate predictability. The work presented here also stands out as an application of advanced tools to the analysis of empirical, real-world data.

Influence of autocorrelation on the topology of the climate network

Different definitions of links in climate networks may lead to considerably different network topologies. We construct a network from climate records of surface level atmospheric temperature in different geographical sites around the globe using two commonly used definitions of links. Utilizing detrended fluctuation analysis, shuffled surrogates, and separation analysis of maritime and continental records, we find that one of the major influences on the structure of climate networks is due to the autocorrelation in the records, which may introduce spurious links. This may explain why different methods could lead to different climate network topologies.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

A Gaussian graphical model approach to climate networks

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyze networks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations. Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.

Networks from flows--from dynamics to topology

Complex network approaches have recently been applied to continuous spatial dynamical systems, like climate, successfully uncovering the system's interaction structure. However the relationship between the underlying atmospheric or oceanic flow's dynamics and the estimated network measures have remained largely unclear. We bridge this crucial gap in a bottom-up approach and define a continuous analytical analogue of Pearson correlation networks for advection-diffusion dynamics on a background flow. Analysing complex networks of prototypical flows and from time series data of the equatorial Pacific, we find that our analytical model reproduces the most salient features of these networks and thus provides a general foundation of climate networks. The relationships we obtain between velocity field and network measures show that line-like structures of high betweenness mark transition zones in the flow rather than, as previously thought, the propagation of dynamical information.

Very early warning of next El Niño

The most important driver of climate variability is the El Niño Southern Oscillation, which can trigger disasters in various parts of the globe. Despite its importance, conventional forecasting is still limited to 6 mo ahead. Recently, we developed an approach based on network analysis, which allows projection of an El Niño event about 1 y ahead. Here we show that our method correctly predicted the absence of El Niño events in 2012 and 2013 and now announce that our approach indicated (in September 2013 already) the return of El Niño in late 2014 with a 3-in-4 likelihood. We also discuss the relevance of the next El Niño to the question of global warming and the present hiatus in the global mean surface temperature.

Disentangling different types of El Niño episodes by evolving climate network analysis

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 for surface air temperatures. We provide a detailed study of the temporal variability of several global climate network characteristics. Based on a simple conceptual view of 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 Niño 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 Niño Southern Oscillation.

Dominant imprint of Rossby waves in the climate network

The connectivity pattern of networks based on ground level temperature records shows a dense stripe of links in the extra tropics of the southern hemisphere. We show that statistical categorization of these links yields a clear association with the pattern of an atmospheric Rossby wave, one of the major mechanisms associated with the weather system and with planetary scale energy transport. It is shown that alternating densities of negative and positive links are arranged in half Rossby wave distances around 3500, 7000, and 10 000 km and are aligned with the expected direction of energy flow, distribution of time delays, and the seasonality of these waves. In addition, long distance links that are associated with Rossby waves are the most dominant in the climate network.

Climate dynamics: a network-based approach for the analysis of global precipitation

Precipitation is one of the most important meteorological variables for defining the climate dynamics, but the spatial patterns of precipitation have not been fully investigated yet. The complex network theory, which provides a robust tool to investigate the statistical interdependence of many interacting elements, is used here to analyze the spatial dynamics of annual precipitation over seventy years (1941-2010). The precipitation network is built associating a node to a geographical region, which has a temporal distribution of precipitation, and identifying possible links among nodes through the correlation function. The precipitation network reveals significant spatial variability with barely connected regions, as Eastern China and Japan, and highly connected regions, such as the African Sahel, Eastern Australia and, to a lesser extent, Northern Europe. Sahel and Eastern Australia are remarkably dry regions, where low amounts of rainfall are uniformly distributed on continental scales and small-scale extreme events are rare. As a consequence, the precipitation gradient is low, making these regions well connected on a large spatial scale. On the contrary, the Asiatic South-East is often reached by extreme events such as monsoons, tropical cyclones and heat waves, which can all contribute to reduce the correlation to the short-range scale only. Some patterns emerging between mid-latitude and tropical regions suggest a possible impact of the propagation of planetary waves on precipitation at a global scale. Other links can be qualitatively associated to the atmospheric and oceanic circulation. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken. The African Sahel, Eastern Australia and Northern Europe regions again appear as the supernodes of the network, confirming furthermore their long-range connection structure. Almost all North-American and Asian nodes vanish, revealing that extreme events can enhance high precipitation gradients, leading to a systematic absence of long-range patterns.

Improved El Nino forecasting by cooperativity detection

Although anomalous episodic warming of the eastern equatorial Pacific, dubbed El Niño by Peruvian fishermen, has major (and occasionally devastating) impacts around the globe, robust forecasting is still limited to about 6 mo ahead. A significant extension of the prewarning time would be instrumental for avoiding some of the worst damages such as harvest failures in developing countries. Here we introduce a unique avenue toward El Niño prediction based on network methods, inspecting emerging teleconnections. Our approach starts from the evidence that a large-scale cooperative mode--linking the El Niño basin (equatorial Pacific corridor) and the rest of the ocean--builds up in the calendar year before the warming event. On this basis, we can develop an efficient 12-mo forecasting scheme, i.e., achieve some doubling of the early-warning period. Our method is based on high-quality observational data available since 1950 and yields hit rates above 0.5, whereas false-alarm rates are below 0.1.

Stability of climate networks with time

The pattern of local daily fluctuations of climate fields such as temperatures and geopotential heights is not stable and hard to predict. Surprisingly, we find that the observed relations between such fluctuations in different geographical regions yields a very robust network pattern that remains highly stable during time. Using a new systematic methodology we track the origins of the network stability. It is found that about half of this network stability is due to the spatial 2D embedding of the network, and half is due to physical coupling between climate in different locations. We also find that around the equator, the contribution of the physical coupling is significantly less pronounced compared to off–equatorial regimes. Finally, we show that there is a gradual monotonic modification of the network pattern as a function of altitude difference.

Scaling of seismic memory with earthquake size

It has been observed that discrete earthquake events possess memory, i.e., that events occurring in a particular location are dependent on the history of that location. We conduct an analysis to see whether continuous real-time data also display a similar memory and, if so, whether such autocorrelations depend on the size of earthquakes within close spatiotemporal proximity. We analyze the seismic wave form database recorded by 64 stations in Japan, including the 2011 "Great East Japan Earthquake," one of the five most powerful earthquakes ever recorded, which resulted in a tsunami and devastating nuclear accidents. We explore the question of seismic memory through use of mean conditional intervals and detrended fluctuation analysis (DFA). We find that the wave form sign series show power-law anticorrelations while the interval series show power-law correlations. We find size dependence in earthquake autocorrelations: as the earthquake size increases, both of these correlation behaviors strengthen. We also find that the DFA scaling exponent α has no dependence on the earthquake hypocenter depth or epicentral distance.

Emergence of El Niño as an autonomous component in the climate network

We construct and analyze a climate network which represents the interdependent structure of the climate in different geographical zones and find that the network responds in a unique way to El Niño events. Analyzing the dynamics of the climate network shows that when El Niño events begin, the El Niño basin partially loses its influence on its surroundings. After typically three months, this influence is restored while the basin loses almost all dependence on its surroundings and becomes autonomous. The formation of an autonomous basin is the missing link to understand the seemingly contradicting phenomena of the afore-noticed weakening of the interdependencies in the climate network during El Niño and the known impact of the anomalies inside the El Niño basin on the global climate system.

Unraveling spurious properties of interaction networks with tailored random networks

We investigate interaction networks that we derive from multivariate time series with methods frequently employed in diverse scientific fields such as biology, quantitative finance, physics, earth and climate sciences, and the neurosciences. Mimicking experimental situations, we generate time series with finite length and varying frequency content but from independent stochastic processes. Using the correlation coefficient and the maximum cross-correlation, we estimate interdependencies between these time series. With clustering coefficient and average shortest path length, we observe unweighted interaction networks, derived via thresholding the values of interdependence, to possess non-trivial topologies as compared to Erdös-Rényi networks, which would indicate small-world characteristics. These topologies reflect the mostly unavoidable finiteness of the data, which limits the reliability of typically used estimators of signal interdependence. We propose random networks that are tailored to the way interaction networks are derived from empirical data. Through an exemplary investigation of multichannel electroencephalographic recordings of epileptic seizures--known for their complex spatial and temporal dynamics--we show that such random networks help to distinguish network properties of interdependence structures related to seizure dynamics from those spuriously induced by the applied methods of analysis.

Coupling detrended fluctuation analysis for analyzing coupled nonstationary signals

When many variables are coupled to each other, a single case study could not give us thorough and precise information. When these time series are stationary, different methods of random matrix analysis and complex networks can be used. But, in nonstationary cases, the multifractal-detrended-cross-correlation-analysis (MF-DXA) method was introduced for just two coupled time series. In this article, we have extended the MF-DXA to the method of coupling detrended fluctuation analysis (CDFA) for the case when more than two series are correlated to each other. Here, we have calculated the multifractal properties of the coupled time series, and by comparing CDFA results of the original series with those of the shuffled and surrogate series, we can estimate the source of multifractality and the extent to which our series are coupled to each other. We illustrate the method by selected examples from air pollution and foreign exchange rates.

Quantifying and modeling long-range cross correlations in multiple time series with applications to world stock indices

We propose a modified time lag random matrix theory in order to study time-lag cross correlations in multiple time series. We apply the method to 48 world indices, one for each of 48 different countries. We find long-range power-law cross correlations in the absolute values of returns that quantify risk, and find that they decay much more slowly than cross correlations between the returns. The magnitude of the cross correlations constitutes "bad news" for international investment managers who may believe that risk is reduced by diversifying across countries. We find that when a market shock is transmitted around the world, the risk decays very slowly. We explain these time-lag cross correlations by introducing a global factor model (GFM) in which all index returns fluctuate in response to a single global factor. For each pair of individual time series of returns, the cross correlations between returns (or magnitudes) can be modeled with the autocorrelations of the global factor returns (or magnitudes). We estimate the global factor using principal component analysis, which minimizes the variance of the residuals after removing the global trend. Using random matrix theory, a significant fraction of the world index cross correlations can be explained by the global factor, which supports the utility of the GFM. We demonstrate applications of the GFM in forecasting risks at the world level, and in finding uncorrelated individual indices. We find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices, which is relevant information for world managers in reducing their portfolio risk. Finally, we argue that this general method can be applied to a wide range of phenomena in which time series are measured, ranging from seismology and physiology to atmospheric geophysics.

Inferring long memory processes in the climate network via ordinal pattern analysis

We use ordinal patterns and symbolic analysis to construct global climate networks and uncover long- and short-term memory processes. Data analyzed are the monthly averaged surface air temperature (SAT field), and the results suggest that the time variability of the SAT field is determined by patterns of oscillatory behavior that repeat from time to time, with a periodicity related to intraseasonal oscillations and to El Niño on seasonal-to-interannual time scales.

An exploration of climate data using complex networks

Climate change is a pressing focus of research, social and economic concern, and political attention. According to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), increased frequency of extreme events will only intensify the occurrence of natural hazards, affecting global population, health, and economies. It is of keen interest to identify "regions" of similar climatological behavior to discover spatial relationships in climate variables, including long-range teleconnections. To that end, we consider a complex networks-based representation of climate data. Cross correlation is used to weight network edges, thus respecting the temporal nature of the data, and a community detection algorithm identifies multivariate clusters. Examining networks for consecutive periods allows us to study structural changes over time. We show that communities have a climatological interpretation and that disturbances in structure can be an indicator of climate events (or lackthereof). Finally, we discuss how this model can be applied for the discovery of more complex concepts such as unknown teleconnections or the development of multivariate climate indices and predictive insights.

Network inference with confidence from multivariate time series

Networks--collections of interacting elements or nodes--abound in the natural and manmade worlds. For many networks, complex spatiotemporal dynamics stem from patterns of physical interactions unknown to us. To infer these interactions, it is common to include edges between those nodes whose time series exhibit sufficient functional connectivity, typically defined as a measure of coupling exceeding a predetermined threshold. However, when uncertainty exists in the original network measurements, uncertainty in the inferred network is likely, and hence a statistical propagation of error is needed. In this manuscript, we describe a principled and systematic procedure for the inference of functional connectivity networks from multivariate time series data. Our procedure yields as output both the inferred network and a quantification of uncertainty of the most fundamental interest: uncertainty in the number of edges. To illustrate this approach, we apply a measure of linear coupling to simulated data and electrocorticogram data recorded from a human subject during an epileptic seizure. We demonstrate that the procedure is accurate and robust in both the determination of edges and the reporting of uncertainty associated with that determination.

Statistical analysis of the overnight and daytime return

We investigate the two components of the total daily return (close-to-close), the overnight return (close-to-open), and the daytime return (open-to-close), as well as the corresponding volatilities of the 2215 New York Stock Exchange stocks for the 20 year period from 1988 to 2007. The tail distribution of the volatility, the long-term memory in the sequence, and the cross correlation between different returns are analyzed. Our results suggest that (i) the two component returns and volatilities have features similar to that of the total return and volatility. The tail distribution follows a power law for all volatilities, and long-term correlations exist in the volatility sequences but not in the return sequences. (ii) The daytime return contributes more to the total return. Both the tail distribution and the long-term memory of the daytime volatility are more similar to that of the total volatility, compared to the overnight records. In addition, the cross correlation between the daytime return and the total return is also stronger. (iii) The two component returns tend to be anticorrelated. Moreover, we find that the cross correlations between the three different returns (total, overnight, and daytime) are quite stable over the entire 20 year period.