# 2016

### Edge anisotropy and the geometric perspective on flow networks

Authors: Nora Molkenthin, Hannes Kutza, Liubov Tupikina, Norbert Marwan, Jonathan F. Donges, Ulrike Feudel, Jürgen Kurths, Reik V. Donner

### Abstract

Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding existing spatial constraints), this work takes a geometric perspective, which considers vertices and edges as objects in a metric space and quantifies the corresponding spatial distribution and alignment. For this purpose, we introduce the concept of edge anisotropy and define a class of measures characterizing the spatial directedness of connections. Specifically, we demonstrate that the local anisotropy of edges incident to a given vertex provides useful information about the local geometry of geophysical flows based on networks constructed from spatio-temporal data, which is complementary to topological characteristics of the same flow networks. Taken both structural and geometric viewpoints together can thus assist the identification of underlying flow structures from observations of scalar variables.

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### Correlation networks from flows. The case of forced and time-dependent advection-diffusion dynamics

Authors: Liubov Tupikina, Nora Molkenthin, Cristóbal López, Emilio Hernández-García, Norbert Marwan, Jürgen Kurths

### Abstract

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.

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### Testing reanalysis datasets in Antarctica: Trends, persistence properties and trend significance

Authors: Yang WangDong Zhou,  Armin Bunde, Shlomo Havlin

### Abstract

The reanalysis datasets provide very important sources for investigating the climate dynamics and climate changes in Antarctica. In this paper, three major reanalysis data are compared with Antarctic station data over the last 35 years: the National Centers for Environmental Prediction and the National Center for Atmospheric Research reanalysis (NCEP1), NCEP-DOE Reanalysis 2 (NCEP2), and the European Centre for Medium-Range Weather Forecasts Interim Re-Analysis (ERA-Interim). In our assessment, we compare the linear trends, the fluctuations around the trends, the persistence properties and the significance level of warming trends in the reanalysis data with the observational ones. We find that NCEP1 and NCEP2 show spurious warming trends in all parts of Antarctica except the Peninsula, while ERA-Interim is quite reliable except at Amundsen-Scott. To investigate the persistence of the data sets, we consider the lag-1 autocorrelation $C(1)$ and the Hurst exponent. While $C(1)$ varies quite erratically in different stations, the Hurst exponent shows similar patterns all over Antarctica. Regarding the significance of the trends, NCEP1 and NCEP2 differ considerably from the observational datasets by strongly exaggerating the warming trends. In contrast, ERA-Interim gives reliable results at most stations except at Amundsen-Scott where it shows a significant cooling trend.

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### Oceanic El-Niño wave dynamics and climate networks

Authors: Yang Wang, Avi Gozolchiani, Yosef Ashkenazy, Shlomo Havlin

Published in: ,

### Abstract

The so-called El Niño-southern oscillation (ENSO) is the most important and influential climate phenomenon of contemporary climate variability, in which oceanic wave dynamics plays an important role. Here we develop and apply an approach based on network theory to quantify the characteristics of El-Niño related oceanic waves using the satellite dataset. We associate the majority of dominant long distance (?500 km) links of the network with several kinds of oceanic waves, i.e. equatorial Kelvin, Rossby, and tropical instability waves. Notably, we find that the location of the out-going () and in-coming hubs () of the climate network coincide with the locations of the wave initiation and dissipation, respectively. We also find that this dissipation at  is much weaker during El-Niño times. Moreover, the hubs of the equatorial network agree with the locations of westerly wind burst activity and high wind vorticity, two mechanisms that were associated with Rossby waves activity. This novel quantification method that is directly based on observational data leads to a better understanding of the oceanic wave dynamics, and it can also improve our understanding of El-Niño dynamics or its prediction.

### A Climate Network Based Stability Index for El Niño Variability

Authors: Qing Yi Feng and Henk A. Dijkstra

Published in arXiv:1503.05449v1  [physics.ao-ph]   (2016)

### Abstract

Most of the existing prediction methods gave a false alarm regarding the El  Niño event in 2014 [1]. A crucial aspect is currently limiting the success of such predictions [2, 3, 4], i.e. the stability of the slowly varying Paci?c climate. This property determines whether sea surface temperature perturbations will be ampli?ed by coupled ocean-atmosphere feedbacks or not. The so-called Bjerknes stability index [5, 6, 7, 8] has been developed for this purpose, but its evaluation is severely constrained by data availability. Here we present a new promising background stability index based on complex network theory [9, 10, 11]. This index e?ciently monitors the changes in spatial correlations in the Paci?c climate and can be evaluated by using only sea surface temperature data.

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### Linking basin-scale connectivity, oceanography and population dynamics for the conservation and management of marine ecosystems

Authors: Mélodie Dubois, Vincent Rossi, Enrico Ser-Giacomi, Sophie Arnaud-Haond, Cristóbal López and Emilio Hernández-García

published in Global Ecology and Biogeography (2016)

### Abstract

Assessing the spatial structure and dynamics of marine populations is still a major challenge in ecology. The need to manage marine resources from ecosystem and large-scale perspectives is recognized, but our partial understanding of oceanic connectivity limits the implementation of globally pertinent conservation planning. Based on a biophysical model for the entire Mediterranean Sea, this study takes an ecosystem approach to connectivity and provides a systematic characterization of broad-scale larval dispersal patterns. It builds on our knowledge of population dynamics and discusses the ecological and management implications.

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### Interdecadal Variability of Southeastern South America Rainfall and Moisture Sources during the Austral Summertime

Authors:  Verónica Martín-Gómez,   Emilio Hernandez-Garcia, Marcelo Barreiro, Cristobal Lopez

in Journal of Climate (2015) (submitted, pending)

### Abstract

This  study  focuses  on  Southeastern  South  America  (SESA),  a  region  that  covers  Uruguay  and  portions  of  northeastern  Argentina  and  South  Brazil  (see  Table  1  and  16 Figure 1). SESA corresponds mostly to the southern part (south of 25S) of the La Plata Basin (LPB), the second largest basin in South America which  comprehends parts of Brazil,  Paraguay,  Uruguay, Argentina  and  Bolivia.  SESA,  located  to  the  south  of  the Amazon  basin,  is  one  of  the  most  densely  populated  regions  in  South  America.  Precipitation and its variability is very important over the region because it plays a key  role  in  the  generation  of  hydroelectric  energy  and  in  the  economy,  which  is  mainly  based on harvesting and ranching (Berbery and Barros 2002). Moisture that could lead to future precipitations over the region can come from two different sources: (i) water vapor advection from others regions, and/or (ii) local recycling.

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### Percolation-based precursors of transitions in spatially extended systems

Authors: Victor Rodriguez-Mendez, Victor M. Eguiluz, Emilio Hernandez-Garcia, Jose J. Ramasco

in  arXiv.org > cond-mat > arXiv:1601.01978  (2016)

### Abstract

Abrupt transitions are ubiquitous in the dynamics of complex systems. Finding early indicators of their arrival, precursors, is fundamental in many areas of science such as ecology, electrical engineering, physiology or climate. However, obtaining warnings of an approaching transition well in advance remains an elusive task. Here we show that a functional network, constructed from spatial correlations of the system's time series, experiences a percolation transition way before the actual system reaches a bifurcation point due to the collective phenomena leading to the global change. Concepts from percolation theory are then used to introduce early warning precursors that anticipate the system's tipping point. We illustrate the generality and versatility of our percolation-based framework with model systems experiencing different types of bifurcations and with Sea Surface Temperature time series associated to El Nino phenomenon.

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# 2015

### Teleconnection Paths via Climate Network Direct Link Detection

Authors: Dong Zhou, Avi Gozolchiani, Yosef Ashkenazy, and Shlomo Havlin

in Phys. Rev. Lett. 115, 268501 – 30 December 2015

### Abstract

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.

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### Analysis of oceans’ influence on spring time rainfall variabilityover Southeastern South America during the 20th century

Authors:  Verónica Martín-Gómez,   Marcelo Barreiro

in Int. J. Climatol. (2015)

### Abstract

Southeastern South America (SESA) rainfall is influenced by the tropical Paciic, Atlantic and Indian Oceans. At the same time, these tropical oceans interact with each other inducing sea surface temperature anomalies in remote basins through atmospheric and oceanic teleconnections. In this study, we employ a tool from complex networks to analyse the collective influence of the three  tropical oceans on austral spring rainfall variability over SESA during the 20th century. To do so we construct a climate network considering as nodes the observed Niño3.4, Tropical North Atlantic (TNA), and Indian Ocean Dipole (IOD) indices, together with an observed and simulated precipitation (PCP) index over SESA. The mean network distance is considered as a measure of synchronization among all these phenomena during the 20th century. The approach allowed to uncover two main synchronization periods characterized by different interactions among the oceanic and precipitation nodes. Whereas in the 1930s El Niño and the TNA were the main tropical oceanic phenomena that inluenced SESA precipitation variability, during the 1970s they were El Niño and the IOD. The influence of El Niño on SESA precipitation variability might be understood through an increase of the northerly transport of moisture in lower-levels and advection of cyclonic vorticity in upper-levels. On the other hand, the interaction between the IOD and PCP can be interpreted in two possible ways. One possibility is that both nodes (IOD and PCP) are forced by El Niño. Another possibility is that the Indian Ocean warming influences rainfall over SESA through the eastward propagation of Rossby waves as suggested previously. Finally, the influence of TNA on SESA precipitation persists even when the El Niño signal is removed, suggesting that SST anomalies in the TNA can directly influence SESA precipitation and further studies are needed to elucidate this connection.

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

Authors: J. F. Donges, J. Heitzig,  B. Beronov, M. Wiedermann, J. RungeQing Yi FengLiubov TupikinaVeronika Stolbova, R. V. Donner, N Marwan, H. A. Dijkstra and J. Kurths

Published in: Chaos 25, 113101 (2015)

### Abstract

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.

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### Information Recovery in Behavioral Networks

in PLoS ONE 10(5): e0125077. (2015)

### Abstract

In the context of agent based modeling and network theory, we focus on the problem of recovering behavior-related choice information from origin-destination type data, a topic also known under the name of network tomography. As a basis for predicting agents' choices we emphasize the connection between adaptive intelligent behavior, causal entropy maximization, and self-organized behavior in an open dynamic system. We cast this problem in the form of binary and weighted networks and suggest information theoretic entropy-driven methods to recover estimates of the unknown behavioral flow parameters. Our objective is to recover the unknown behavioral values across the ensemble analytically, without explicitly sampling the configuration space. In order to do so, we consider the Cressie-Read family of entropic functionals, enlarging the set of estimators commonly employed to make optimal use of the available information. More specifically, we explicitly work out two cases of particular interest: Shannon functional and the likelihood functional. We then employ them for the analysis of both univariate and bivariate data sets, comparing their accuracy in reproducing the observed trends.

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### Par@Graph – a parallel toolbox for the construction and analysis of large complex climate networks

Authors:  H. IhshaishA. Tantet, J. C. M. Dijkzeul, and H. A. Dijkstra

in Geosci. Model Dev., 8, 3321-3331, 2015

### Abstract

In this paper, we present Par@Graph, a software toolbox to reconstruct and analyze complex climate networks having a large number of nodes (up to at least 106) and edges (up to at least 1012). The key innovation is an efficient set of parallel software tools designed to leverage the inherited hybrid parallelism in distributed-memory clusters of multi-core machines. The performance of the toolbox is illustrated through networks derived from sea surface height (SSH) data of a global high-resolution ocean model. Less than 8 min are needed on 90 Intel Xeon E5-4650 processors to reconstruct a climate network including the preprocessing and the correlation of 3 × 105 SSH time series, resulting in a weighted graph with the same number of vertices and about 3.2 × 108 edges. In less than 14 min on 30 processors, the resulted graph's degree centrality, strength, connected components, eigenvector centrality, entropy and clustering coefficient metrics were obtained. These results indicate that a complete cycle to construct and analyze a large-scale climate network is available under 22 min Par@Graph therefore facilitates the application of climate network analysis on high-resolution observations and model results, by enabling fast network reconstruct from the calculation of statistical similarities between climate time series. It also enables network analysis at unprecedented scales on a variety of different sizes of input data sets.

### The Construction of Complex Networks from Linear and Nonlinear Measures – Climate Networks

Authors:   J. Ignacio Deza, Hisham Ihshaish

in http://www.sciencedirect.com/science/journal/18770509/51/supp/CProcedia Computer Science, Volume 51, 2015, Pages 404-412, ISSN 1877-0509

### Abstract

During the last decade the techniques of complex network analysis have found application in climate research. The main idea consists in embedding the  characteristics of climate variables, e.g., temperature, pressure or rainfall, into the topology of complex networks by appropriate linear and nonlinear measures. Applying such measures on climate time series leads to defining links between their corresponding locations on the studied region, whereas the locations are the network's nodes. The resulted networks, consequently, are analysed using the various network analysis tools present in literature in order to get a better insight on the processes, patterns and interactions occurring in climate system. In this regard we present ClimNet; a complete set of software tools to construct climate networks based on a wide range of linear (cross correlation) and nonlinear (Information theoretic) measures. The presented software will allow the construction of large networks’ adjacency matrices from climate time series while supporting functions to tune relationships to different time-scales by means of symbolic ordinal analysis. The provided tools have been used in the production of various original contributions in climate research. This work presents an in-depth description of the implemented statistical functions widely used to construct climate networks. Additionally, a general overview of the architecture of the developed software is provided as well as a brief analysis of application examples.

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### Dominant transport pathways in an atmospheric blocking event

Authors:  Enrico Ser-GiacomiRuggero Vasile, Irene Recuerda, Emilio Hernández-García and Cristóbal López

in Chaos 25, 087413 (2015)

### Abstract

A Lagrangian flow network is constructed for the atmospheric blocking of Eastern Europe and Western Russia in summer 2010. We compute the most probable paths followed by fluid particles, which reveal the Omega-block skeleton of the event. A hierarchy of sets of highly probable paths is introduced to describe transport pathways when the most probable path alone is not representative enough. These sets of paths have the shape of narrow coherent tubes flowing close to the most probable one. Thus, even when the most probable path is not very significant in terms of its probability, it still identifies the geometry of the transport pathways.

### A study of the air–sea interaction in the South Atlantic Convergence Zone through Granger causality

Authors: Giulio Tirabassi, Cristina Masoller and Marcelo Barreiro

Published in the International  Journal of Climatology, Volume 35, Issue 12

October 2015, Pages 3440–3453

### Abstract

Air–sea interaction in the region of the South Atlantic Convergence Zone (SACZ) is studied using Granger causality (GC) as a measure of directional coupling. Calculation of the area weighted connectivity indicates that the SACZ region is the one with largest mutual air–sea connectivity in the south Atlantic basin during summertime. Focusing on the leading mode of daily coupled variability, GC allows distinguishing four regimes characterized by different coupling: there are years in which the forcing is mainly directed from the atmosphere to the ocean, years in which the ocean forces the atmosphere, years in which the influence is mutual and years in which the coupling is not significant. A composite analysis shows that ocean-driven events have atmospheric anomalies that develop first and are strongest over the ocean, while in events without coupling anomalies develop from the continent where they are strongest and have smaller oceanic extension.

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

Authors: J. Ignacio Deza, Marcelo Barreiro, Cristina Masoller

Published  in:  Chaos 25, 033105 (2015)

### Abstract

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. Two data-sets 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 extratropical 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 extra-tropics 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.

### Flow networks: A characterization of geophysical fluid transport

Authors:  Enrico Ser-Giacomi, Vincent Rossi, Cristóbal López and Emilio Hernández-García

in Chaos 25, 036404 (2015)

### Abstract

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 networknodes. 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 networkinto coherent regions, i.e., areas internally well mixed, but with little fluid interchange between them.

### Inferring the connectivity of coupled oscillators from time-series statistical similarity analysis

Authors: Giulio Tirabassi, Ricardo Sevilla-Escoboza, Javier M. Buldú,  Cristina Masoller

Published in: Scientific Reports 5, Article number: 10829 (2015)

### Abstract

A system composed by interacting dynamical elements can be represented by a network, where the nodes represent the elements that constitute the system, and the links account for their interactions, which arise due to a variety of mechanisms, and which are often unknown. A popular method for inferring the system connectivity (i.e., the set of links among pairs of nodes) is by performing a statistical similarity analysis of the time-series collected from the dynamics of the nodes. Here, by considering two systems of coupled oscillators (Kuramoto phase oscillators and Rössler chaotic electronic oscillators) with known and controllable coupling conditions, we aim at testing the performance of this inference method, by using linear and non linear statistical similarity measures. We find that, under adequate conditions, the network links can be perfectly inferred, i.e., no mistakes are made regarding the presence or absence of links. These conditions for perfect inference require: i) an appropriated choice of the observed variable to be analysed, ii) an appropriated interaction strength, and iii) an adequate thresholding of the similarity matrix. For the dynamical units considered here we find that the linear statistical similarity measure performs, in general, better than the non-linear ones.

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### An early warning indicator for atmospheric blocking events using transfer operators

Authors:  Alexis Tantet, Fiona R. van der Burgt, Henk A. Dijkstra

Published in: Chaos 25, 036406 (2015)

### Abstract

The existence of persistent midlatitude atmospheric flow regimes with time-scales larger than 5-10 days and indications of preferred transitions between them motivates to develop early warning indicators for such regime transitions. In this paper, we use a hemispheric barotropic model together with estimates of transfer operators on a reduced phase space to develop an early warning indicator of the zonal to blocked flow transition in this model. It is shown that, the spectrum of the transfer operators can be used to study the slow dynamics of the flow as well as the non-Markovian character of the reduction. The slowest motions are thereby found to have time scales of three to six weeks and to be associated with meta-stable regimes (and their transitions) which can be detected as almost-invariant sets of the transfer operator. From the energy budget of the model, we are able to explain the meta-stability of the regimes and the existence of preferred transition paths. Even though the model is highly simplified, the skill of the early warning indicator is promising, suggesting that the transfer operator approach can be used in parallel to an operational deterministic model for stochastic prediction or to assess forecast uncertainty.

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

Authors: Enrico Ser-GiacomiRuggero Vasile, Emilio Hernández-García and Cristóbal López;

in  Phys. Rev. E 92, 012818 (July 2015)

### Abstract

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 which 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.

### Closed-loop separation control using machine learning

Authors: N. Gautier, J.-L. Aider, T. Duriez, B. R. NoackM. Segond and M. Abel

Published in: Journal of Fluid Mechanics, Volume 770, May 2015, pp 442- 457

### Abstract

We present the first closed-loop separation control experiment using a novel, model-free strategy based on genetic programming, which we call ‘machine learning control’. The goal is to reduce the recirculation zone of backward-facing step flow at manipulated by a slotted jet and optically sensed by online particle image velocimetry. The feedback control law is optimized with respect to a cost functional based on the recirculation area and a penalization of the actuation. This optimization is performed employing genetic programming. After 12 generations comprised of 500 individuals, the algorithm converges to a feedback law which reduces the recirculation zone by 80 %. This machine learning control is benchmarked against the best periodic forcing which excites Kelvin–Helmholtz vortices. The machine learning control yields a new actuation mechanism resonating with the low-frequency flapping mode instability. This feedback control performs similarly to periodic forcing at the design condition but outperforms periodic forcing when the Reynolds number is varied by a factor two. The current study indicates that machine learning control can effectively explore and optimize new feedback actuation mechanisms in numerous experimental applications.

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# 2014

### Evolution of atmospheric connectivity in the 20th century

Authors:  F. Arizmendi, A. C. Martí, and M. Barreiro

in Nonlinear Processes in Geophysics, Volume 21, Issue 4, 2014, pp.825-839,

### Abstract

We aim to study the evolution of the upper atmosphere connectivity over the 20th century as well as to distinguish the oceanically forced component from the atmospheric internal variability. For this purpose we build networks from two different reanalysis data sets using both linear and nonlinear statistical similarity measures to determine the existence of links between different regions of the world in the two halves of the last century. We furthermore use symbolic analysis to emphasize intra-seasonal, intra-annual and inter-annual timescales. Both linear and nonlinear networks have similar structures and evolution, showing that the most connected regions are in the tropics over the Pacific Ocean. Also, the Southern Hemisphere extratropics have more connectivity in the first half of the 20th century, particularly on intra-annual and intra-seasonal timescales.

Changes over the Pacific main connectivity regions are analyzed in more detail. Both linear and nonlinear networks show that the central and western Pacific regions have decreasing connectivity from early 1900 up to about 1940, when it starts increasing again until the present. The inter-annual network shows a similar behavior. However, this is not true of other timescales. On intra-annual timescales the minimum connectivity is around 1956, with a negative (positive) trend before (after) that date for both the central and western Pacific. While this is also true of the central Pacific on intra-seasonal timescales, the western Pacific shows a positive trend during the entire 20th century.

In order to separate the internal and forced connectivity networks and to study their evolution through time, an ensemble of atmospheric general circulation model outputs is used. The results suggest that the main connectivity patterns captured in the reanalysis networks are due to the oceanically forced component, particularly on inter-annual timescales. Moreover, the atmospheric internal variability seems to play an important role in determining the intra-seasonal timescale networks.

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### Interaction network based early-warning indicators of vegetation transitions

Authors: Giulio Tirabassi, J. Viebahn, V. Dakos, H.A. Dijkstra, C. Masoller, M. Rietkerk, S.C. Dekker

in Ecological Complexity 19, 148-157 (2014)

### Abstract

Changes in vegetation patterns in semi-arid regions can precede the abrupt transition to bare soil. Here, complex network techniques are used to develop novel early-warning indicators for these desertification transitions. These indicators are applied to results from a local positive feedback vegetation model and are compared to classical indicators, such as the autocorrelation and variance of biomass time series. A quantitative measure is also introduced to evaluate the quality of the early-warning indicators. Based on this measure, the network-based indicators are superior to the classical ones, being more sensitive to the presence of the transition point.

### Deep ocean early warning signals of an Atlantic MOC collapse

Authors: Q. Y. Feng, J. P. Viebahn, and H. Dijkstra

in Geophys. Res. Lett., 41, 6009–6015

### Abstract

A future collapse of the Atlantic Meridional Overturning Circulation (MOC) has been identified as one of the most dangerous tipping points in the climate system. It is therefore crucial to develop early warning indicators for such a potential collapse based on relatively short time series. So far, attempts to use indicators based on critical slowdown have been marginally successful. Based on complex climate network reconstruction, we here present a promising new indicator for the MOC collapse that efficiently monitors spatial changes in deep ocean circulation. Through our analysis of the performance of this indicator, we formulate optimal locations of measurement of the MOC to provide early warning signals of a collapse. Our results imply that an increase in spatial resolution of the Atlantic MOC observations (i.e., at more sections) can improve early detection, because the spatial coherence in the deep ocean arising near the transition is better captured.

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### Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka

Authors: V. Stolbova,  P. Martin, B. Bookhagen, N. Marwan, and J. Kurths

in Nonlin. Processes Geophys., 21, 901–917 (2014)

### Abstract

This paper employs a complex network approach to determine the topology and evolution of the network of extreme precipitation that governs the organization of extreme rainfall before, during, and after the Indian Summer Monsoon (ISM) season. We construct networks of extreme rainfall events during the ISM (June–September), post-monsoon (October–December), and pre-monsoon (March–May) periods from satellite-derived (Tropical Rainfall Measurement Mission, TRMM) and rain-gauge interpolated (Asian Precipitation Highly Resolved Observational Data Integration Towards the Evaluation of Water Resources, APHRODITE) data sets. The structure of the networks is determined by the level of synchronization of extreme rainfall events between different grid cells throughout the Indian subcontinent. Through the analysis of various complex-network metrics, we describe typical repetitive patterns in North Pakistan (NP), the Eastern Ghats (EG), and the Tibetan Plateau (TP). These patterns appear during the pre-monsoon season, evolve during the ISM, and disappear during the post-monsoon season. These are important meteorological features that need further attention and that may be useful in ISM timing and strength prediction.

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### On the influence of spatial sampling on climate networks

Authors: N. Molkenthin , K. Rehfeld , V. Stolbova, , L. Tupikina , and J. Kurths

in Nonlin. Processes Geophys., 21, 651-657 (2014)

### Abstract

Climate networks are constructed from climate time series data using correlation measures. It is widely accepted that the geographical proximity, as well as other geographical features such as ocean and atmospheric currents, have a large impact on the observable time-series similarity. Therefore it is to be expected that the spatial sampling will influence the reconstructed network. Here we investigate this by comparing analytical flow networks, networks generated with the START model and networks from temperature data from the Asian monsoon domain. We evaluate them on a regular grid, a grid with added random jittering and two variations of clustered sampling. We find that the impact of the spatial sampling on most network measures only distorts the plots if the node distribution is significantly inhomogeneous. As a simple diagnostic measure for the detection of inhomogeneous sampling we suggest the Voronoi cell size distribution.

### Characterizing the evolution of climate networks

Authors: L. Tupikina  , K. Rehfeld , N. Molkenthin , V. Stolbova , N. Marwan, and J. Kurths

in Nonlin. Processes Geophys., 21, 705-711 (2014)

### Abstract

Complex network theory has been successfully applied to understand the structural and functional topology of many dynamical systems from nature, society and technology. Many properties of these systems change over time, and, consequently, networks reconstructed from them will, too. However, although static and temporally changing networks have been studied extensively, methods to quantify their robustness as they evolve in time are lacking. In this paper we develop a theory to investigate how networks are changing within time based on the quantitative analysis of dissimilarities in the network structure.

Our main result is the common component evolution function (CCEF) which characterizes network development over time. To test our approach we apply it to several model systems, Erd?s–Rényi networks, analytically derived flow-based networks, and transient simulations from the START model for which we control the change of single parameters over time. Then we construct annual climate networks from NCEP/NCAR reanalysis data for the Asian monsoon domain for the time period of 1970–2011 CE and use the CCEF to characterize the temporal evolution in this region. While this real-world CCEF displays a high degree of network persistence over large time lags, there are distinct time periods when common links break down. This phasing of these events coincides with years of strong El Niño/Southern Oscillation phenomena, confirming previous studies. The proposed method can be applied for any type of evolving network where the link but not the node set is changing, and may be particularly useful to characterize nonstationary evolving systems using complex networks.

### Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves

Authors: Vincent Rossi; Enrico Ser-Giacomi; Cristobal Lopez; Emilio Hernandez-Garcia

in Geophysical Research Letters 41, 2883-2891 (2014)

### Abstract

Oceanic dispersal and connectivity have been identified as crucial factors for structuring marine populations and designing Marine Protected Areas (MPAs). Focusing on larval dispersal by ocean currents, we propose an approach coupling Lagrangian transport and new tools from Network Theory to characterize marine connectivity in the Mediterranean basin. Larvae of different pelagic durations and seasons are modeled as passive tracers advected in a simulated oceanic surface flow from which a network of connected areas is constructed. Hydrodynamical provinces extracted from this network are delimited by frontiers which match multi-scale oceanographic features. By examining the repeated occurrence of such boundaries, we identify the spatial scales and geographic structures that would control larval dispersal across the entire seascape. Based on these hydrodynamical units, we study novel connectivity metrics for existing reserves. Our results are discussed in the context of ocean biogeography and MPAs design, having ecological and managerial implications.

### Are North Atlantic multidecadal SST anomalies westward propagating?

Authors: Q. Y. Feng and H. Dijkstra

in Geophysical Research Letters, Article first published online: 21 Jan 2014

### Abstract

The westward propagation of sea surface temperature (SST) anomalies is one of the main characteristics of one of the theories of the Atlantic Multidecadal Oscillation. Here we use techniques from complex network modeling to investigate the existence of the westward propagation in the North Atlantic SST observations. We construct Climate Networks (CNs) by using a linear Pearson correlation measure (resulting in Pearson Correlation Climate Network (PCCNs)) and a (nonlinear) mutual information measure (resulting in Mutual Information Climate Network (MICNs)) of spatial correlations between SST variations. Analysis of the topological properties of both types of CNs shows that the MICNs are better in capturing the main features of propagating patterns from the noisy SST data than PCCNs and that westward propagation of multidecadal SST anomalies indeed seems to occur in the North Atlantic.

### Simultaneous first- and second-order percolation transitions in interdependent networks

Authors: D. Zhou, A. Bashan, R. Cohen, Y. Berezin, N. Shnerb and S. Havlin

in  Phys. Rev. E 90, 012803 – Published 8 July 2014

### Abstract

In a system of interdependent networks, an initial failure of nodes invokes a cascade of iterative failures that may lead to a total collapse of the whole system in the form of an abrupt first-order transition. When the fraction of initial failed nodes 1-p reaches criticality p=pc, the abrupt collapse occurs by spontaneous cascading failures. At this stage, the giant component decreases slowly in a plateau form and the number of iterations in the cascade ? diverges. The origin of this plateau and its increasing with the size of the system have been unclear. Here we find that, simultaneously with the abrupt first-order transition, a spontaneous second-order percolation occurs during the cascade of iterative failures. This sheds light on the origin of the plateau and how its length scales with the size of the system. Understanding the critical nature of the dynamical process of cascading failures may be useful for designing strategies for preventing and mitigating catastrophic collapses.

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### Distinguishing the effects of internal and forced atmospheric variability in climate networks

Author: J. I. Deza, C. Masoller, and M. Barreiro

Published in : Nonlin. Processes Geophys., 21, 617-631, 2014

### Abstract

The fact that the climate on the earth is a highly complex dynamical system is well-known. In the last few decades great deal of effort has been focused on understanding how climate phenomena in one geographical region affects the climate of other regions. Complex networks are a powerful framework for identifying climate interdependencies. To further exploit the knowledge of the links uncovered via the network analysis (for, e.g., improvements in prediction), a good understanding of the physical mechanisms underlying these links is required. Here we focus on understanding the role of atmospheric variability, and construct climate networks representing internal and forced variability using the output of an ensemble of AGCM runs. A main strength of our work is that we construct the networks using MIOP (mutual information computed from ordinal patterns), which allows the separation of intraseasonal, intra-annual and interannual timescales. This gives further insight to the analysis of climatological data. The connectivity of these networks allows us to assess the influence of two main indices, NINO3.4 – one of the indices used to describe ENSO (El Niño–Southern oscillation) – and of the North Atlantic Oscillation (NAO), by calculating the networks from time series where these indices were linearly removed. A main result of our analysis is that the connectivity of the forced variability network is heavily affected by "El Niño": removing the NINO3.4 index yields a general loss of connectivity; even teleconnections between regions far away from the equatorial Pacific Ocean are lost, suggesting that these regions are not directly linked, but rather, are indirectly interconnected via El Niño, particularly at interannual timescales. On the contrary, on the internal variability network – independent of sea surface temperature (SST) forcing – the links connecting the Labrador Sea with the rest of the world are found to be significantly affected by NAO, with a maximum at intra-annual timescales. While the strongest non-local links found are those forced by the ocean, the presence of teleconnections due to internal atmospheric variability is also shown.

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### An interaction network perspective on the relation between patterns of sea surface temperature variability and global mean surface temperature

Authors: A. Tantet and H. A. Dijkstra

in Earth Syst. Dynam., 5, 1-14, 2014

### Abstract

On interannual-to-multidecadal time scales variability in sea surface temperature appears to be organized in large-scale spatiotemporal patterns. In this paper, we investigate these patterns by studying the community structure of interaction networks constructed from sea surface temperature observations. Much of the community structure as well as the first neighbour maps can be interpreted using known dominant patterns of variability, such as the El Niño/Southern Oscillation and the Atlantic Multidecadal Oscillation and teleconnections. The community detection method allows to overcome some shortcomings of Empirical Orthogonal Function analysis or composite analysis and hence provides additional information with respect to these classical analysis tools. The community analysis provides also new insight into the relationship between patterns of sea surface temperature and the global mean surface temperature (GMST). On the decadal-to-multidecadal time scale, we show that only two communities (Indian Ocean and North Atlantic) determine most of the GMST variability.

# 2013

### Percolation of interdependent networks with intersimilarity

Authors:  Yanqing Hu, Dong Zhou, Rui Zhang, Zhangang Han, Céline Rozenblat, and Shlomo Havlin

Published in Phys. Rev. E 88, 052805, 7 November 2013

### ABSTRACT

Real data show that interdependent networks usually involve intersimilarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent [Parshani et al. Europhys. Lett. 92, 68002 (2010)]. For example, the coupled worldwide port network and the global airport network are intersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines, exist in both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboring nodes in one network depend on neighboring nodes in the other network, we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results of Parshani et al. suggest that intersimilarity has considerable effects on reducing the cascading failures; however, a theoretical understanding of this effect on the cascading process is currently missing. Here we map the cascading process with intersimilarity to a percolation of networks composed of components of common links and noncommon links. This transforms the percolation of intersimilar system to a regular percolation on a series of subnetworks, which can be solved analytically. We apply our analysis to the case where the network of common links is an Erd?s-Rényi (ER) network with the average degree K, and the two networks of noncommon links are also ER networks. We show for a fully coupled pair of ER networks, that for any K?0, although the cascade is reduced with increasing K, the phase transition is still discontinuous. Our analysis can be generalized to any kind of interdependent random network systems.

### Global climate network evolves with North Atlantic Oscillation phases: Coupling to Southern Pacific Ocean

Authors: O. Guez, A. Gozolchiani, Y. Berezin, Y. Wang and S. Havlin

in ,  (Oct 2013)

### Abstract

We construct a network from climate records of the atmospheric temperature at the surface level, at different geographical sites in the globe, using reanalysis data from years 1948–2010. We find that the network correlates with the North Atlantic Oscillation (NAO), both locally in the North Atlantic, and through coupling to the Southern Pacific Ocean. The existence of tele-connection links between those areas and their stability over time allows us to suggest a possible physical explanation for this phenomenon.

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### Dominant Imprint of Rossby Waves in the Climate Network

Authors: Y. Wang, A. Gozolchiani, Y. Ashkenazy, Y. Berezin, O. Guez, S. Havlin

in Physical Review Letters 111, 138501 (2013)

### Abstract

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 10000 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.

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### Interaction network based early warning indicators for the Atlantic MOC collapse

Authors: M. van der Mheen, M., H. A. Dijkstra, A. Gozolchiani, M. den Toom, Q.Y. Feng, J. Kurths, and E. Hernandez-Garcia

in Geophysical Research Letters, Volume 40, Issue 11, pages 2714–2719 (2013)

### Abstract

Early warning indicators of the collapse of the Atlantic Meridional Overturning Circulation (MOC) have up to now mostly been based on temporal correlations in single time series. Here, we propose new indicators based on spatial correlations in the time series of the Atlantic temperature field. To demonstrate the performance of these indicators, we use a meridional depth model of the MOC for which the critical conditions for collapse can be explicitly computed. An interaction network approach is used to monitor changes in spatial correlations in the model temperature time series as the critical transition is approached. The new early warning indicators are based on changes in topological properties of the network, in particular changes in the distribution functions of the degree and the clustering coefficient.

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### Inferring interdependencies in climate networks constructed at inter-annual, intra-season and longer time scales

Authors: J.I. Deza, M. Barreiro and C. Masoller

in Eur. Phys. J. Special Topics 222 , 511–523 (2013)

### Abstract

We study global climate networks constructed by means of ordinal time series analysis. Climate interdependencies among the nodes are quantified by the mutual information, computed from time series of monthly-averaged surface air temperature anomalies, and from their symbolic ordinal representation (OP). This analysis allows identifying topological changes in the network when varying the time-interval of the ordinal pattern. We consider intra-season time-intervals (e.g., the patterns are formed by anomalies in consecutive months) and inter-annual time-intervals (e.g., the patterns are formed by anomalies in consecutive years). We discuss how the network density and topology change with these time scales, and provide evidence of correlations between geographically distant regions that occur at specific time scales. In particular, we find that an increase in the ordinal pattern spacing (i.e., an increase in the timescale of the ordinal analysis), results in climate networks with increased connectivity on the equatorial Pacific area. On the contrary, the number of significant links decreases when the ordinal analysis is done with a shorter timescale (by comparing consecutive months), and interpret this effect as due to more stochasticity in the time-series in the short timescale. As the equatorial Pacific is known to be dominated by El Ni˜no-Southern Oscillation (ENSO) on scales longer than several months, our methodology allows constructing climate networks where the effect of ENSO goes from mild (monthly OP) to intense (yearly OP), independently of the length of the ordinal pattern and of the thresholding method employed.

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### On the effects of lag-times in networks constructed from similarities of monthlyfluctuations of climate fields

Authors: G. Tirabassi and C. Masoller

in EPL, 102 (2013) 59003

### Abstract

The complex network framework has been successfully applied to the analysis of climatological data, providing, for example, a better understanding of the mechanisms underlying reduced predictability during El Nino or La Nina years. Despite the large interest that climate networks have attracted, several issues remain to be investigated. Here we focus on the influence of the periodic solar forcing in climate networks constructed via similarities of monthly averaged Surface Air Temperature (SAT) anomalies. We shift the time series in each pair of nodes such as to superpose their seasonal cycles. In this way, when two nodes are located in different hemispheres we are able to quantify the similarity of SAT anomalies during the winters and during the summers. We find that data time-shifting does not significantly modify the network Area Weighted Connectivity (AWC), which is the fraction of the total area of the Earth to which each node is connected. This unexpected network property can be understood in terms of how data time-shifting modifies the strength of the links connecting geographical regions in different hemispheres, and how these modifcations are washed out by averaging the AWC.

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