Climate networks around the globe are significantly affected by El Niño

Community structure of tropics emerging from spatio-temporal variations in the Intertropical Convergence Zone dynamics

The Intertropical Convergence Zone (ITCZ) is a narrow tropical belt of deep convective clouds, intense precipitation, and monsoon circulations encircling the Earth. Complex interactions between the ITCZ and local geophysical dynamics result in high climate variability, making weather forecasting and prediction of extreme rainfall or drought events challenging. We unravel the complex spatio-temporal dynamics of the ITCZ and the resulting teleconnection patterns via a novel tropical climate classification achieved using complex network analysis and community detection. We reduce the high-dimensional complex ITCZ dynamics into a simple yet insightful community structure that classifies the tropics into seven regions representing distinct ITCZ dynamics. The two largest communities, encompassing landmasses over the Northern and Southern hemispheres, are associated with coherent seasonal ITCZ dynamics and have significant long-range connections. Temporal analysis of the community structure highlights that the tropical Pacific and Atlantic Oceans communities exhibit substantial variation on multidecadal scales. Further, these communities exhibit incoherent dynamics due to atmosphere-ocean interactions driven by equatorial and coastal oceanic upwelling.

Multiple serial correlations in global air temperature anomaly time series

Serial correlations within temperature time series serve as indicators of the temporal consistency of climate events. This study delves into the serial correlations embedded in global surface air temperature (SAT) data. Initially, we preprocess the SAT time series to eradicate seasonal patterns and linear trends, resulting in the SAT anomaly time series, which encapsulates the inherent variability of Earth's climate system. Employing diverse statistical techniques, we identify three distinct types of serial correlations: short-term, long-term, and nonlinear. To identify short-term correlations, we utilize the first-order autoregressive model, AR(1), revealing a global pattern that can be partially attributed to atmospheric Rossby waves in extratropical regions and the Eastern Pacific warm pool. For long-term correlations, we adopt the standard detrended fluctuation analysis, finding that the global pattern aligns with long-term climate variability, such as the El Niño-Southern Oscillation (ENSO) over the Eastern Pacific. Furthermore, we apply the horizontal visibility graph (HVG) algorithm to transform the SAT anomaly time series into complex networks. The topological parameters of these networks aptly capture the long-term correlations present in the data. Additionally, we introduce a novel topological parameter, Δσ, to detect nonlinear correlations. The statistical significance of this parameter is rigorously tested using the Monte Carlo method, simulating fractional Brownian motion and fractional Gaussian noise processes with a predefined DFA exponent to estimate confidence intervals. In conclusion, serial correlations are universal in global SAT time series and the presence of these serial correlations should be considered carefully in climate sciences.

Climate network analysis of extreme events: Tropical cyclones

Climate networks based on surface air temperature data are analyzed to identify distinct signatures of tropical cyclones, which appear in the Indian Ocean. These networks, which are percolating networks, show an abrupt phase transition in the order parameter and the susceptibility during cyclonic events. The behavior seen is compared for the months October-November 2016, when three successive cyclones, viz., cyclone Kyant, cyclone Nada, and cyclone Vardah, were seen, and compared with a year where a single cyclone, cyclone Ockhi, was seen in December 2017. All these cyclones were seen in the Bay of Bengal. The microtransitions, i.e., the locations of jumps in the order parameter, for these two cases show distinct patterns. The signatures of the cyclones can be seen in other quantities like the node degrees and their geographic distributions and other network characterizers. We also compare these with a cyclone, cyclone Ashoba (2015), seen in the Arabian Sea where cyclones are rarer. The networks also show the signatures of precursor behavior, which has implications for further analysis.

Exploring the intensity, distribution and evolution of teleconnections using climate network analysis

Teleconnections refer to long-range climate system linkages occurring over typically thousands of kilometers. Generally speaking, most teleconnections are attributed to the transmission of energy and propagation of waves although the physical complexity and characteristics behind these waves are not fully understood. To address this knowledge gap, we develop a climate network-based approach to reveal their directions and distribution patterns, evaluate the intensity of teleconnections, and identify sensitive regions using global daily surface air temperature data. Our results reveal a stable average intensity distribution pattern for teleconnections across a substantial spatiotemporal scale from 1948 to 2021, with the extent and intensity of teleconnection impacts increasing more prominently in the Southern Hemisphere over the past 37 years. Furthermore, we pinpoint climate-sensitive regions, such as southeastern Australia, which are likely to face increasing impacts due to global warming. Our proposed method offers new insights into the dynamics of global climate patterns and can inform strategies to address climate change and extreme events.

Signatures of climatic phenomena in climate networks: El Niño and La Niña

We construct climate networks based on surface air temperature data to identify distinct signatures of climatic phenomena such as El Niño and La Niña events, which trigger many climatic disruptions around the globe with severe economic and ecological consequences. The climate network has been seen to show a discontinuous phase transition in the size of the normalized largest cluster and the susceptibility during both events. The correlation matrix of the network shows a structure that has distinct characteristics for El Niño events, La Niña events, and normal conditions of the Pacific Ocean. We also identify the signatures of the El Niño southern oscillations in the heat map of the cross-correlation and network quantifiers like the betweenness centrality. The distribution of teleconnections of distinct strengths, the betweenness centrality distributions, and the geographic location of nodes of high betweenness centrality yield important insights into the structure of the network and the transfer of information between different parts. We further discuss the predictive power of these quantities.

A modified extreme event-based synchronicity measure for climate time series

Extreme event-based synchronicity is a specific measure of similarity of extreme event-like time series. It is capable to capture the nonlinear interactions between climatic extreme events. In this study, we proposed a modified extreme event-based synchronicity measure that incorporates two types of extreme events (positive and negative) simultaneously in climate anomalies to characterize the synchronization and time delays. Statistical significance of the modified extreme event synchronization measure is tested by Monte-Carlo simulations. The applications of the modified extreme event-based synchronicity measure on synthetic time series verified that it was superior to the traditional event synchronicity measure. Both synchronous and antisynchronous features between climate time series could be captured by the modified measure. It is potentially applied in investigating the interrelationship between climate extremes and climate index or constructing complex networks of climate variables. In addition, this modified extreme event-based synchronicity measure could be easily applied to other types of time series just by identifying the extreme events properly.

Feature learning and network structure from noisy node activity data

In the studies of network structures, much attention has been devoted to developing approaches to reconstruct networks and predict missing links when edge-related information is given. However, such approaches are not applicable when we are only given noisy node activity data with missing values. This work presents an unsupervised learning framework to learn node vectors and construct networks from such node activity data. First, we design a scheme to generate random node sequences from node context sets, which are generated from node activity data. Then, a three-layer neural network is adopted training the node sequences to obtain node vectors, which allow us to construct networks and capture nodes with synergistic roles. Furthermore, we present an entropy-based approach to select the most meaningful neighbors for each node in the resulting network. Finally, the effectiveness of the method is validated through both synthetic and real data.

Partial event coincidence analysis for distinguishing direct and indirect coupling in functional network construction

Correctly identifying interaction patterns from multivariate time series presents an important step in functional network construction. In this context, the widespread use of bivariate statistical association measures often results in a false identification of links because strong similarity between two time series can also emerge without the presence of a direct interaction due to intermediate mediators or common drivers. In order to properly distinguish such direct and indirect links for the special case of event-like data, we present here a new generalization of event coincidence analysis to a partial version thereof, which is aimed at excluding possible transitive effects of indirect couplings. Using coupled chaotic systems and stochastic processes on two generic coupling topologies (star and chain configuration), we demonstrate that the proposed methodology allows for the correct identification of indirect interactions. Subsequently, we apply our partial event coincidence analysis to multi-channel EEG recordings to investigate possible differences in coordinated alpha band activity among macroscopic brain regions in resting states with eyes open (EO) and closed (EC) conditions. Specifically, we find that direct connections typically correspond to close spatial neighbors while indirect ones often reflect longer-distance connections mediated via other brain regions. In the EC state, connections in the frontal parts of the brain are enhanced as compared to the EO state, while the opposite applies to the posterior regions. In general, our approach leads to a significant reduction in the number of indirect connections and thereby contributes to a better understanding of the alpha band desynchronization phenomenon in the EO state.

A global [Formula: see text] gene co-expression network constructed from hundreds of experimental conditions with missing values

BACKGROUND

Gene co-expression networks (GCNs) can be used to determine gene regulation and attribute gene function to biological processes. Different high throughput technologies, including one and two-channel microarrays and RNA-sequencing, allow evaluating thousands of gene expression data simultaneously, but these methodologies provide results that cannot be directly compared. Thus, it is complex to analyze co-expression relations between genes, especially when there are missing values arising for experimental reasons. Networks are a helpful tool for studying gene co-expression, where nodes represent genes and edges represent co-expression of pairs of genes.

RESULTS

In this paper, we establish a method for constructing a gene co-expression network for the Anopheles gambiae transcriptome from 257 unique studies obtained with different methodologies and experimental designs. We introduce the sliding threshold approach to select node pairs with high Pearson correlation coefficients. The resulting network, which we name AgGCN1.0, is robust to random removal of conditions and has similar characteristics to small-world and scale-free networks. Analysis of network sub-graphs revealed that the core is largely comprised of genes that encode components of the mitochondrial respiratory chain and the ribosome, while different communities are enriched for genes involved in distinct biological processes.

CONCLUSION

Analysis of the network reveals that both the architecture of the core sub-network and the network communities are based on gene function, supporting the power of the proposed method for GCN construction. Application of network science methodology reveals that the overall network structure is driven to maximize the integration of essential cellular functions, possibly allowing the flexibility to add novel functions.

Network-based forecasting of climate phenomena

Network theory, as emerging from complex systems science, can provide critical predictive power for mitigating the global warming crisis and other societal challenges. Here we discuss the main differences of this approach to classical numerical modeling and highlight several cases where the network approach substantially improved the prediction of high-impact phenomena: 1) El Niño events, 2) droughts in the central Amazon, 3) extreme rainfall in the eastern Central Andes, 4) the Indian summer monsoon, and 5) extreme stratospheric polar vortex states that influence the occurrence of wintertime cold spells in northern Eurasia. In this perspective, we argue that network-based approaches can gainfully complement numerical modeling.

Eigen microstates and their evolution of global ozone at different geopotential heights

Studies on stratospheric ozone have attracted much attention due to its serious impacts on climate changes and its important role as a tracer of Earth's global circulation. Tropospheric ozone as a main atmospheric pollutant damages human health as well as the growth of vegetation. Yet, there is still a lack of a theoretical framework to fully describe the variation of ozone. To understand ozone's spatiotemporal variance, we introduce the eigen microstate method to analyze the global ozone mass mixing ratio between January 1, 1979 and June 30, 2020 at 37 pressure layers. We find that eigen microstates at different geopotential heights can capture different climate phenomena and modes. Without deseasonalization, the first eigen microstates capture the seasonal effect and reveal that the phase of the intra-annual cycle moves with the geopotential heights. After deseasonalization, by contrast, the collective patterns from the overall trend, El Niño-Southern Oscillation (ENSO), quasi-biennial oscillation, and tropopause pressure are identified by the first few significant eigen microstates. The theoretical framework proposed here can also be applied to other complex Earth systems.

Precursors of the El Niño phenomenon: A climate network analysis

The identification of precursors of climatic phenomena has enormous practical importance. Recent work constructs a climate network based on surface air temperature data to analyze the El Niño phenomena. We utilize microtransitions which occur before the discontinuous percolation transition in the network as well as other network quantities to identify a set of reliable precursors of El Niño episodes. These precursors identify 10 out of 13 El Niño episodes occurring in the period of 1979-2018 with an average lead time of approximately 6.4 months. We also find indicators of tipping events in the data.

Climate network approach reveals the modes of CO2 concentration to surface air temperature

Increasing atmospheric carbon dioxide (CO₂) is expected to be the main factor of global warming. The relation between CO₂ concentrations and surface air temperature (SAT) has been found related to Rossby waves based on a multi-layer complex network approach. However, the significant relations between CO₂ and SAT occur in the South Hemisphere that is not that much influenced by human activities may offer not enough information to formulate targeted carbon reduction policies. Here, we address it by removing the effects of the Rossby waves to reconstruct CO₂ concentrations and SAT multi-layer complex network. We uncover that the CO₂ concentrations are strongly associated with the surrounding SAT regions. The influential regions of CO₂ on SAT occur over eastern Asia, West Asia, North Africa, the coast of North American, and Western Europe. It is shown that CO₂ over Siberia in phase with the SAT variability in eastern East Asia. Indeed, CO₂ concentration variability is causing effects on the recent warming of SAT in some middle latitude regions. Furthermore, sensitive parameters that CO₂ impacts SAT of top 15 carbon emissions countries have been identified. These countries are significantly responsible for global warming, giving implications for carbon emissions reductions. The methodology and results presented here not only facilitate further research in regions of increased sensitivity to the warming resulting from CO₂ concentrations but also can formulate strategies and countermeasures for carbon emission and carbon reduction.

Statistical physics approaches to the complex Earth system

Global warming, extreme climate events, earthquakes and their accompanying socioeconomic disasters pose significant risks to humanity. Yet due to the nonlinear feedbacks, multiple interactions and complex structures of the Earth system, the understanding and, in particular, the prediction of such disruptive events represent formidable challenges to both scientific and policy communities. During the past years, the emergence and evolution of Earth system science has attracted much attention and produced new concepts and frameworks. Especially, novel statistical physics and complex networks-based techniques have been developed and implemented to substantially advance our knowledge of the Earth system, including climate extreme events, earthquakes and geological relief features, leading to substantially improved predictive performances. We present here a comprehensive review on the recent scientific progress in the development and application of how combined statistical physics and complex systems science approaches such as critical phenomena, network theory, percolation, tipping points analysis, and entropy can be applied to complex Earth systems. Notably, these integrating tools and approaches provide new insights and perspectives for understanding the dynamics of the Earth systems. The overall aim of this review is to offer readers the knowledge on how statistical physics concepts and theories can be useful in the field of Earth system science.

Measuring topological descriptors of complex networks under uncertainty

Revealing the structural features of a complex system from the observed collective dynamics is a fundamental problem in network science. To compute the various topological descriptors commonly used to characterize the structure of a complex system (e.g., the degree, the clustering coefficient, etc.), it is usually necessary to completely reconstruct the network of relations between the subsystems. Several methods are available to detect the existence of interactions between the nodes of a network. By observing some physical quantities through time, the structural relationships are inferred using various discriminating statistics (e.g., correlations, mutual information, etc.). In this setting, the uncertainty about the existence of the edges is reflected in the uncertainty about the topological descriptors. In this study, we propose a methodological framework to evaluate this uncertainty, replacing the topological descriptors, even at the level of a single node, with appropriate probability distributions, eluding the reconstruction phase. Our theoretical framework agrees with the numerical experiments performed on a large set of synthetic and real-world networks. Our results provide a grounded framework for the analysis and the interpretation of widely used topological descriptors, such as degree centrality, clustering, and clusters, in scenarios in which the existence of network connectivity is statistically inferred or when the probabilities of existence π_{ij} of the edges are known. To this purpose, we also provide a simple and mathematically grounded process to transform the discriminating statistics into the probabilities π_{ij}.

Correlation networks of air particulate matter ( PM 2.5 ): a comparative study

Over the last decades, severe haze pollution constitutes a major source of far-reaching environmental and human health problems. The formation, accumulation and diffusion of pollution particles occurs under complex temporal scales and expands throughout a wide spatial coverage. Seeking to understand the transport patterns of haze pollutants in China, we review a proposed framework of time-evolving directed and weighted air quality correlation networks. In this work, we evaluate monitoring stations' time-series data from China and California, to test the sensitivity of the framework to region size, climate and pollution magnitude across multiple years (2014-2020). We learn that the use of hourly PM 2.5 concentration data is needed to detect periodicities in the positive and negative correlations of the concentrations. In addition, we show that the standardization of the correlation function method is required to obtain networks with more meaningful links when evaluating the dispersion of a severe haze event at the North China Plain or a wildfire event in California during December 2017. Post COVID-19 outbreak in China, we observe a significant drop in the magnitude of the assigned weights, indicating the improved air quality and the slowed transport of PM 2.5 due to the lockdown. To identify regions where pollution transport is persistent, we extend the framework, partitioning the dynamic networks and reducing the networks' complexity through node subsampling. The end result separates the temporal series of PM 2.5 in set of regions that are similarly affected through the year.

Epidemic spreading and control strategies in spatial modular network

Epidemic spread on networks is one of the most studied dynamics in network science and has important implications in real epidemic scenarios. Nonetheless, the dynamics of real epidemics and how it is affected by the underline structure of the infection channels are still not fully understood. Here we apply the susceptible-infected-recovered model and study analytically and numerically the epidemic spread on a recently developed spatial modular model imitating the structure of cities in a country. The model assumes that inside a city the infection channels connect many different locations, while the infection channels between cities are less and usually directly connect only a few nearest neighbor cities in a two-dimensional plane. We find that the model experience two epidemic transitions. The first lower threshold represents a local epidemic spread within a city but not to the entire country and the second higher threshold represents a global epidemic in the entire country. Based on our analytical solution we proposed several control strategies and how to optimize them. We also show that while control strategies can successfully control the disease, early actions are essentials to prevent the disease global spread.

A Novel Information Theoretical Criterion for Climate Network Construction

This paper presents a novel methodology for Climate Network (CN) construction based on the Kullback-Leibler divergence (KLD) among Membership Probability (MP) distributions, obtained from the Second Order Data-Coupled Clustering (SODCC) algorithm. The proposed method is able to obtain CNs with emergent behaviour adapted to the variables being analyzed, and with a low number of spurious or missing links. We evaluate the proposed method in a problem of CN construction to assess differences in wind speed prediction at different wind farms in Spain. The considered problem presents strong local and mesoscale relationships, but low synoptic scale relationships, which have a direct influence in the CN obtained. We carry out a comparison of the proposed approach with a classical correlation-based CN construction method. We show that the proposed approach based on the SODCC algorithm and the KLD constructs CNs with an emergent behaviour according to underlying wind speed prediction data physics, unlike the correlation-based method that produces spurious and missing links. Furthermore, it is shown that the climate network construction method facilitates the evaluation of symmetry properties in the resulting complex networks.

A network-based toolkit for evaluation and intercomparison of weather prediction and climate modeling

Model evaluation is a critical component in the development and applications of environmental modeling systems. Conventional metrics such as Pearson product-moment correlation coefficient (r), root-mean-square error (RMSE), and mean absolute error (MAE), albeit process-based and limited to point-to-point statistical comparison, have been widely used in model evaluations. In this study, we propose a network-based toolkit for evaluation of model performance and multi-model comparisons with applications to weather prediction and climate modeling. The model outputs are topologically quantified through a range of network metrics to provide a holistic measure of system dynamics. We first use this toolkit to evaluate the performance of air temperature simulated by the Weather Research and Forecasting model with station measurements over the contiguous United States (CONUS). Results of network analysis suggest a good match between simulation and measurement, as indicated by conventional metrics (r, RMSE, and MAE) as well. The sensitivity of these network metrics is then analyzed based on CONUS station measurements with additive random errors using Monte Carlo simulations. Network metrics show more sensitive and highly nonlinear responses to increasing random errors as compared to conventional ones. Moreover, we use the new toolkit for intercomparison of the downscaled historical air temperature outputs from four global climate models. The similarity in both metrics and spatial structure highlights the capability of network analysis for capturing system dynamics in models alike. The network theory is therefore promising for evaluation and intercomparison of various environmental modeling systems with complex dynamics.

Synchronization of extreme rainfall during the Australian summer monsoon: Complex network perspectives

Monsoon rains are an important fresh water supply for agricultural activity, while extreme rainfalls during a monsoon season frequently cause flash floods. In this study, a nonlinear causation measure of event synchronization is used to set complex networks of extreme rainfall during the Australian summer monsoon (ASM) development between 1st November and 1st March. We adopted Tropical Rainfall Measuring Mission-based satellite rain rate estimates from 1998 to 2015. Examining several standard network centrality measures, such as degree and local clustering, we revealed the multiscale nature of ASM development, which previously was only studied by weather analysis methods. The land-sea contrast in surface heating critical for ASM is depicted clearly by the degree centrality. In addition, both the clustering coefficient and the community structure show critical change in spatial pattern matching with the climatological average onset time of the ASM during late December. The former is likely related to the interaction between synoptic forcing and mesoscale convection during monsoon onset, resulting in characteristic changes in the rainfall field. One of the network communities also extends spatially during the onset, revealing critical information from the near-equatorial region to ASM and would be applicable to monitor monsoon development. Results from this study further support that network measures as defined by a single parameter of rainfall have enormous potential for monsoon onset prediction.

Interconnections between networks acting like an external field in a first-order percolation transition

Many interdependent, real-world infrastructures involve interconnections between different communities or cities. Here we show how the effects of such interconnections can be described as an external field for interdependent networks experiencing a first-order percolation transition. We find that the critical exponents γ and δ, related to the external field, can also be defined for first-order transitions but that they have different values than those found for second-order transitions. Surprisingly, we find that both sets of different exponents (for first and second order) can even be found within a single model of interdependent networks, depending on the dependency coupling strength. Nevertheless, in both cases both sets satisfy Widom's identity, δ-1=γ/β, which further supports the validity of their definitions. Furthermore, we find that both Erdős-Rényi and scale-free networks have the same values of the exponents in the first-order regime, implying that these models are in the same universality class. In addition, we find that in k-core percolation the values of the critical exponents related to the field are the same as for interdependent networks, suggesting that these systems also belong to the same universality class.

Mapping atmospheric waves and unveiling phase coherent structures in a global surface air temperature reanalysis dataset

In the analysis of empirical signals, detecting correlations that capture genuine interactions between the elements of a complex system is a challenging task with applications across disciplines. Here, we analyze a global dataset of surface air temperature (SAT) with daily resolution. Hilbert analysis is used to obtain phase, instantaneous frequency, and amplitude information of SAT seasonal cycles in different geographical zones. The analysis of the phase dynamics reveals large regions with coherent seasonality. The analysis of the instantaneous frequencies uncovers clean wave patterns formed by alternating regions of negative and positive correlations. In contrast, the analysis of the amplitude dynamics uncovers wave patterns with additional large-scale structures. These structures are interpreted as due to the fact that the amplitude dynamics is affected by processes that act in long and short time scales, while the dynamics of the instantaneous frequency is mainly governed by fast processes. Therefore, Hilbert analysis allows us to disentangle climatic processes and to track planetary atmospheric waves. Our results are relevant for the analysis of complex oscillatory signals because they offer a general strategy for uncovering interactions that act at different time scales.

Generalized early warning signals in multivariate and gridded data with an application to tropical cyclones

Tipping events in dynamical systems have been studied across many applications, often by measuring changes in variance or autocorrelation in a one-dimensional time series. In this paper, methods for detecting early warning signals of tipping events in multidimensional systems are reviewed and expanded. An analytical justification of the use of dimension-reduction by empirical orthogonal functions, in the context of early warning signals, is provided and the one-dimensional techniques are also extended to spatially separated time series over a 2D field. The challenge of predicting an approaching tropical cyclone by a tipping-point analysis of the sea-level pressure series is used as the primary example, and an analytical model of a moving cyclone is also developed in order to test predictions. We show that the one-dimensional power spectrum indicator may be used following dimension-reduction or over a 2D field. We also show the validity of our moving cyclone model with respect to tipping-point indicators.

Community detection analysis in wind speed-monitoring systems using mutual information-based complex network

A mutual information-based weighted network representation of a wide wind speed-monitoring system in Switzerland was analyzed in order to detect communities. Two communities have been revealed, corresponding to two clusters of sensors situated, respectively, on the Alps and on the Jura-Plateau that define the two major climatic zones of Switzerland. The silhouette measure is used to evaluate the obtained communities and confirm the membership of each sensor to its cluster.

Climate network percolation reveals the expansion and weakening of the tropical component under global warming

Global climate warming poses a significant challenge to humanity; it is associated with, e.g., rising sea level and declining Arctic sea ice. Increasing extreme events are also considered to be a result of climate warming, and they may have widespread and diverse effects on health, agriculture, economics, and political conflicts. Still, the detection and quantification of climate change, both in observations and climate models, constitute a main focus of the scientific community. Here, we develop an approach based on network and percolation frameworks to study the impacts of climate changes in the past decades using historical models and reanalysis records, and we analyze the expected upcoming impacts using various future global warming scenarios. We find an abrupt transition during the evolution of the climate network, indicating a consistent poleward expansion of the largest cluster that corresponds to the tropical area, as well as the weakening of the strength of links in the tropic. This is found both in the reanalysis data and in the Coupled Model Intercomparison Project Phase 5 (CMIP5) 21st century climate change simulations. The analysis is based on high-resolution surface (2 m) air temperature field records. We discuss the underlying mechanism for the observed expansion of the tropical cluster and associate it with changes in atmospheric circulation represented by the weakening and expansion of the Hadley cell. Our framework can also be useful for forecasting the extent of the tropical cluster to detect its influence on different areas in response to global warming.

Percolation framework of the Earth's topography

Self-similarity and long-range correlations are the remarkable features of the Earth's surface topography. Here we develop an approach based on percolation theory to study the geometrical features of Earth. Our analysis is based on high-resolution, 1 arc min, ETOPO1 global relief records. We find some evidence for abrupt transitions that occurred during the evolution of the Earth's relief network, indicative of a continental/cluster aggregation. We apply finite-size-scaling analysis based on a coarse-graining procedure to show that the observed transition is most likely discontinuous. Furthermore, we study the percolation on two-dimensional fractional Brownian motion surfaces with Hurst exponent H as a model of long-range correlated topography, which suggests that the long-range correlations may play a key role in the observed discontinuity on Earth. Our framework presented here provides a theoretical model to better understand the geometrical phase transition on Earth, and it also identifies the critical nodes that will be more exposed to global climate change in the Earth's relief network.

Complex networks reveal global pattern of extreme-rainfall teleconnections

Climatic observables are often correlated across long spatial distances, and extreme events, such as heatwaves or floods, are typically assumed to be related to such teleconnections^1,2. Revealing atmospheric teleconnection patterns and understanding their underlying mechanisms is of great importance for weather forecasting in general and extreme-event prediction in particular^3,4, especially considering that the characteristics of extreme events have been suggested to change under ongoing anthropogenic climate change^5-8. Here we reveal the global coupling pattern of extreme-rainfall events by applying complex-network methodology to high-resolution satellite data and introducing a technique that corrects for multiple-comparison bias in functional networks. We find that the distance distribution of significant connections (P < 0.005) around the globe decays according to a power law up to distances of about 2,500 kilometres. For longer distances, the probability of significant connections is much higher than expected from the scaling of the power law. We attribute the shorter, power-law-distributed connections to regional weather systems. The longer, super-power-law-distributed connections form a global rainfall teleconnection pattern that is probably controlled by upper-level Rossby waves. We show that extreme-rainfall events in the monsoon systems of south-central Asia, east Asia and Africa are significantly synchronized. Moreover, we uncover concise links between south-central Asia and the European and North American extratropics, as well as the Southern Hemisphere extratropics. Analysis of the atmospheric conditions that lead to these teleconnections confirms Rossby waves as the physical mechanism underlying these global teleconnection patterns and emphasizes their crucial role in dynamical tropical-extratropical couplings. Our results provide insights into the function of Rossby waves in creating stable, global-scale dependencies of extreme-rainfall events, and into the potential predictability of associated natural hazards.

Evaluation of ENSO simulations in CMIP5 models: A new perspective based on percolation phase transition in complex networks

In this study, the performance of CMIP5 models in simulating the El Niño-Southern Oscillation (ENSO) is evaluated by using a new metric based on percolation theory. The surface air temperatures (SATs) over the tropical Pacific Ocean are constructed as a SAT network, and the nodes within the network are linked if they are highly connected (e.g., high correlations). It has been confirmed from reanalysis datasets that the SAT network undergoes an abrupt percolation phase transition when the influences of the sea surface temperature anomalies (SSTAs) below are strong enough. However, from simulations of the CMIP5 models, most models are found incapable of capturing the observed phase transition at a proper critical point P_c. For the 15 considered models, four even miss the phase transition, indicating that the simulated SAT network is too stable to be significantly changed by the SSTA below. Only four models can be considered cautiously with some skills in simulating the observed phase transition of the SAT network. By comparing the simulated SSTA patterns with the node vulnerabilities, which is the chance of each node being isolated during a ENSO event, we find that the improperly simulated sea-air interactions are responsible for the missing of the observed percolation phase transition. Accordingly, a careful study of the sea-air couplers, as well as the atmospheric components of the CMIP5 models is suggested. Since the percolation phase transition of the SAT network is a useful phenomenon to indicate whether the ENSO impacts can be transferred remotely, it deserves more attention for future model development.

Long-range fluctuations and multifractality in connectivity density time series of a wind speed monitoring network

This paper studies the daily connectivity time series of a wind speed-monitoring network using multifractal detrended fluctuation analysis. It investigates the long-range fluctuation and multifractality in the residuals of the connectivity time series. Our findings reveal that the daily connectivity of the correlation-based network is persistent for any correlation threshold. Further, the multifractality degree is higher for larger absolute values of the correlation threshold.

Constructing regional climate networks in the Amazonia during recent drought events

Climate networks are powerful approaches to disclose tele-connections in climate systems and to predict severe climate events. Here we construct regional climate networks from precipitation data in the Amazonian region and focus on network properties under the recent drought events in 2005 and 2010. Both the networks of the entire Amazon region and the extreme networks resulted from locations severely affected by drought events suggest that network characteristics show slight difference between the two drought events. Based on network degrees of extreme drought events and that without drought conditions, we identify regions of interest that are correlated to longer expected drought period length. Moreover, we show that the spatial correlation length to the regions of interest decayed much faster in 2010 than in 2005, which is because of the dual roles played by both the Pacific and Atlantic oceans. The results suggest that hub nodes in the regional climate network of Amazonia have fewer long-range connections when more severe drought conditions appeared in 2010 than that in 2005.

Integrating networks and comparative genomics reveals retroelement proliferation dynamics in hominid genomes

Retroelements (REs) are mobile DNA sequences that multiply and spread throughout genomes by a copy-and-paste mechanism. These parasitic elements are active in diverse genomes, from yeast to humans, where they promote diversity, cause disease, and accelerate evolution. Because of their high copy number and sequence similarity, studying their activity and tracking their proliferation dynamics is a challenge. It is particularly difficult to pinpoint the few REs in a genome that are still active in the haystack of degenerate and suppressed elements. We develop a computational framework based on network theory that tracks the path of RE proliferation throughout evolution. We analyze SVA (SINE-VNTR-Alu), the youngest RE family in human genomes, to understand RE dynamics across hominids. Integrating comparative genomics and network tools enables us to track the course of SVA proliferation, identify yet unknown active communities, and detect tentative "master REs" that played key roles in SVA propagation, providing strong support for the fundamental "master gene" model of RE proliferation. The method is generic and thus can be applied to REs of any of the thousands of available genomes to identify active RE communities and master REs that were pivotal in the evolution of their host genomes.

ENSO teleconnections in the southern hemisphere: A climate network view

Using functional network analysis, we study the seasonality of atmospheric connectivity and its interannual variability depending on the different phases of the El Niño-Southern Oscillation (ENSO) phenomenon. We find a strong variability of the connectivity on seasonal and interannual time scales both in the tropical and extratropical regions. In particular, there are significant changes in the southern hemisphere extratropical atmospheric connectivity during austral spring within the different stages of ENSO: We find that the connectivity patterns due to stationary Rossby waves differ during El Niño and La Niña, showing a very clear wave train originating close to Australia in the former case, as opposed to La Niña that seems to generate a wave train from the central Pacific. An attempt to understand these differences in terms of changes in the frequency of intraseasonal weather regimes cannot fully explain the differences in connectivity, even though we found the prevalence of different intraseasonal regimes in each phase of ENSO. We conclude that the differential response to extreme phases of ENSO during austral springtime is related to the forcing of waves of different tropical origins.

Percolation Phase Transition of Surface Air Temperature Networks: A new test bed for El Niño/La Niña simulations

In this work, we studied the air-sea interaction over the tropical central eastern Pacific from a new perspective, climate network. The surface air temperatures over the tropical Pacific were constructed as a network, and the nodes within this network were linked if they have a similar temporal varying pattern. Using three different reanalysis datasets, we verified the percolation phase transition. That is, when the influences of El Niño/La Niña are strong enough to isolate more than 48% of the nodes, the network may abruptly be divided into many small pieces, indicating a change of the network state. This phenomenon was reproduced successfully by a coupled general circulation model, Flexible Global Ocean-Atmosphere-Land System Model Spectral Version 2, but another model, Flexible Global Ocean-Atmosphere-Land System Model Grid-point Version 2, failed. As both models have the same oceanic component, but are with different atmospheric components, the improperly used atmospheric component should be responsible for the missing of the percolation phase transition. Considering that this new phenomenon is only recently noticed, current state-of-the-art models may ignore this process and induce unrealistic simulations. Accordingly, percolation phase transition is proposed as a new test bed, which deserves more attention in the future.

Network analysis reveals strongly localized impacts of El Niño

Climatic conditions influence the culture and economy of societies and the performance of economies. Specifically, El Niño as an extreme climate event is known to have notable effects on health, agriculture, industry, and conflict. Here, we construct directed and weighted climate networks based on near-surface air temperature to investigate the global impacts of El Niño and La Niña. We find that regions that are characterized by higher positive/negative network "in"-weighted links are exhibiting stronger correlations with the El Niño basin and are warmer/cooler during El Niño/La Niña periods. In contrast to non-El Niño periods, these stronger in-weighted activities are found to be concentrated in very localized areas, whereas a large fraction of the globe is not influenced by the events. The regions of localized activity vary from one El Niño (La Niña) event to another; still, some El Niño (La Niña) events are more similar to each other. We quantify this similarity using network community structure. The results and methodology reported here may be used to improve the understanding and prediction of El Niño/La Niña events and also may be applied in the investigation of other climate variables.

Identifying large-scale patterns of unpredictability and response to insolation in atmospheric data

Understanding the complex dynamics of the atmosphere is of paramount interest due to its impact in the entire climate system and in human society. Here we focus on identifying, from data, the geographical regions which have similar atmospheric properties. We study surface air temperature (SAT) time series with monthly resolution, recorded at a regular grid covering the Earth surface. We consider two datasets: NCEP CDAS1 and ERA Interim reanalysis. We show that two surprisingly simple measures are able to extract meaningful information: i) the distance between the lagged SAT and the incoming solar radiation and ii) the Shannon entropy of SAT and SAT anomalies. The distance uncovers well-defined spatial patterns formed by regions with similar SAT response to solar forcing while the entropy uncovers regions with similar degree of SAT unpredictability. The entropy analysis also allows identifying regions in which SAT has extreme values. Importantly, we uncover differences between the two datasets which are due to the presence of extreme values in one dataset but not in the other. Our results indicate that the distance and entropy measures can be valuable tools for the study of other climatological variables, for anomaly detection and for performing model inter-comparisons.

Smooth information flow in temperature climate network reflects mass transport

A directed climate network is constructed by Granger causality analysis of air temperature time series from a regular grid covering the whole Earth. Using winner-takes-all network thresholding approach, a structure of a smooth information flow is revealed, hidden to previous studies. The relevance of this observation is confirmed by comparison with the air mass transfer defined by the wind field. Their close relation illustrates that although the information transferred due to the causal influence is not a physical quantity, the information transfer is tied to the transfer of mass and energy.

A perturbation-theoretic approach to Lagrangian flow networks

Complex network approaches have been successfully applied for studying transport processes in complex systems ranging from road, railway, or airline infrastructures 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.

Percolation framework to describe El Niño conditions

Complex networks have been used intensively to investigate the flow and dynamics of many natural systems including the climate system. Here, we develop a percolation based measure, the order parameter, to study and quantify climate networks. We find that abrupt transitions of the order parameter usually occur ∼1 year before El Niño events, suggesting that they can be used as early warning precursors of El Niño. Using this method, we analyze several reanalysis datasets and show the potential for good forecasting of El Niño. The percolation based order parameter exhibits discontinuous features, indicating a possible relation to the first order phase transition mechanism.

Clustering coefficient and periodic orbits in flow networks

We show that the clustering coefficient, a standard measure in network theory, when applied to flow networks, i.e., graph representations of fluid flows in which links between nodes represent fluid transport between spatial regions, identifies approximate locations of periodic trajectories in the flow system. This is true for steady flows and for periodic ones in which the time interval τ used to construct the network is the period of the flow or a multiple of it. In other situations, the clustering coefficient still identifies cyclic motion between regions of the fluid. Besides the fluid context, these ideas apply equally well to general dynamical systems. By varying the value of τ used to construct the network, a kind of spectroscopy can be performed so that the observation of high values of mean clustering at a value of τ reveals the presence of periodic orbits of period 3τ, which impact phase space significantly. These results are illustrated with examples of increasing complexity, namely, a steady and a periodically perturbed model two-dimensional fluid flow, the three-dimensional Lorenz system, and the turbulent surface flow obtained from a numerical model of circulation in the Mediterranean sea.

A complex network representation of wind flows

Climate networks have proven to be a valuable method to investigate spatial connectivity patterns of the climate system. However, so far such networks have mostly been applied to scalar observables. In this study, we propose a new method for constructing networks from atmospheric wind fields on two-dimensional isobaric surfaces. By connecting nodes along a spatial environment based on the local wind flow, we derive a network representation of the low-level circulation that captures its most important characteristics. In our approach, network links are placed according to a suitable statistical null model that takes into account the direction and magnitude of the flow. We compare a simulation-based (numerically costly) and a semi-analytical (numerically cheaper) approach to determine the statistical significance of possible connections, and find that both methods yield qualitatively similar results. As an application, we choose the regional climate system of South America and focus on the monsoon season in austral summer. Monsoon systems are generally characterized by substantial changes in the large-scale wind directions, and therefore provide ideal applications for the proposed wind networks. Based on these networks, we are able to reveal the key features of the low-level circulation of the South American Monsoon System, including the South American Low-Level Jet. Networks of the dry and the wet season are compared with each other and their differences are consistent with the literature on South American climate.

Introduction to Focus Issue: Complex network perspectives on flow systems

During the last few years, complex network approaches have demonstrated their great potentials as versatile tools for exploring the structural as well as dynamical properties of dynamical systems from a variety of different fields. Among others, recent successful examples include (i) functional (correlation) network approaches to infer hidden statistical interrelationships between macroscopic regions of the human brain or the Earth's climate system, (ii) Lagrangian flow networks allowing to trace dynamically relevant fluid-flow structures in atmosphere, ocean or, more general, the phase space of complex systems, and (iii) time series networks unveiling fundamental organization principles of dynamical systems. In this spirit, complex network approaches have proven useful for data-driven learning of dynamical processes (like those acting within and between sub-components of the Earth's climate system) that are hidden to other analysis techniques. This Focus Issue presents a collection of contributions addressing the description of flows and associated transport processes from the network point of view and its relationship to other approaches which deal with fluid transport and mixing and/or use complex network techniques.

A generalization of random matrix theory and its application to statistical physics

To study the statistical structure of crosscorrelations in empirical data, we generalize random matrix theory and propose a new method of cross-correlation analysis, known as autoregressive random matrix theory (ARRMT). ARRMT takes into account the influence of auto-correlations in the study of cross-correlations in multiple time series. We first analytically and numerically determine how auto-correlations affect the eigenvalue distribution of the correlation matrix. Then we introduce ARRMT with a detailed procedure of how to implement the method. Finally, we illustrate the method using two examples taken from inflation rates for air pressure data for 95 US cities.

Unravelling the community structure of the climate system by using lags and symbolic time-series analysis

Many natural systems can be represented by complex networks of dynamical units with modular structure in the form of communities of densely interconnected nodes. Unraveling this community structure from observed data requires the development of appropriate tools, particularly when the nodes are embedded in a regular space grid and the datasets are short and noisy. Here we propose two methods to identify communities, and validate them with the analysis of climate datasets recorded at a regular grid of geographical locations covering the Earth surface. By identifying mutual lags among time-series recorded at different grid points, and by applying symbolic time-series analysis, we are able to extract meaningful regional communities, which can be interpreted in terms of large-scale climate phenomena. The methods proposed here are valuable tools for the study of other systems represented by networks of dynamical units, allowing the identification of communities, through time-series analysis of the observed output signals.

Percolation Phase Transition of Surface Air Temperature Networks under Attacks of El Niño/La Niña

In this study, sea surface air temperature over the Pacific is constructed as a network, and the influences of sea surface temperature anomaly in the tropical central eastern Pacific (El Niño/La Niña) are regarded as a kind of natural attack on the network. The results show that El Niño/La Niña leads an abrupt percolation phase transition on the climate networks from stable to unstable or metastable phase state, corresponding to the fact that the climate condition changes from normal to abnormal significantly during El Niño/La Niña. By simulating three different forms of attacks on an idealized network, including Most connected Attack (MA), Localized Attack (LA) and Random Attack (RA), we found that both MA and LA lead to stepwise phase transitions, while RA leads to a second-order phase transition. It is found that most attacks due to El Niño/La Niña are close to the combination of MA and LA, and a percolation critical threshold Pc can be estimated to determine whether the percolation phase transition happens. Therefore, the findings in this study may renew our understandings of the influence of El Niño/La Niña on climate, and further help us in better predicting the subsequent events triggered by El Niño/La Niña.

Correlation Networks from Flows. The Case of Forced and Time-Dependent Advection-Diffusion Dynamics

Complex network theory provides an elegant and powerful framework to statistically investigate different types of systems such as society, brain or the structure of local and long-range dynamical interrelationships in the climate system. Network links in climate networks typically imply information, mass or energy exchange. However, the specific connection between oceanic or atmospheric flows and the climate network's structure is still unclear. We propose a theoretical approach for verifying relations between the correlation matrix and the climate network measures, generalizing previous studies and overcoming the restriction to stationary flows. Our methods are developed for correlations of a scalar quantity (temperature, for example) which satisfies an advection-diffusion dynamics in the presence of forcing and dissipation. Our approach reveals that correlation networks are not sensitive to steady sources and sinks and the profound impact of the signal decay rate on the network topology. We illustrate our results with calculations of degree and clustering for a meandering flow resembling a geophysical ocean jet.

Constructing networks from a dynamical system perspective for multivariate nonlinear time series

We describe a method for constructing networks for multivariate nonlinear time series. We approach the interaction between the various scalar time series from a deterministic dynamical system perspective and provide a generic and algorithmic test for whether the interaction between two measured time series is statistically significant. The method can be applied even when the data exhibit no obvious qualitative similarity: a situation in which the naive method utilizing the cross correlation function directly cannot correctly identify connectivity. To establish the connectivity between nodes we apply the previously proposed small-shuffle surrogate (SSS) method, which can investigate whether there are correlation structures in short-term variabilities (irregular fluctuations) between two data sets from the viewpoint of deterministic dynamical systems. The procedure to construct networks based on this idea is composed of three steps: (i) each time series is considered as a basic node of a network, (ii) the SSS method is applied to verify the connectivity between each pair of time series taken from the whole multivariate time series, and (iii) the pair of nodes is connected with an undirected edge when the null hypothesis cannot be rejected. The network constructed by the proposed method indicates the intrinsic (essential) connectivity of the elements included in the system or the underlying (assumed) system. The method is demonstrated for numerical data sets generated by known systems and applied to several experimental time series.