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  • 1. Barfuss, Wolfram
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Jürgen
    Deterministic limit of temporal difference reinforcement learning for stochastic games2019In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 99, no 4, article id 043305Article in journal (Refereed)
    Abstract [en]

    Reinforcement learning in multiagent systems has been studied in the fields of economic game theory, artificial intelligence, and statistical physics by developing an analytical understanding of the learning dynamics (often in relation to the replicator dynamics of evolutionary game theory). However, the majority of these analytical studies focuses on repeated normal form games, which only have a single environmental state. Environmental dynamics, i.e., changes in the state of an environment affecting the agents' payoffs has received less attention, lacking a universal method to obtain deterministic equations from established multistate reinforcement learning algorithms. In this work we present a methodological extension, separating the interaction from the adaptation timescale, to derive the deterministic limit of a general class of reinforcement learning algorithms, called temporal difference learning. This form of learning is equipped to function in more realistic multistate environments by using the estimated value of future environmental states to adapt the agent's behavior. We demonstrate the potential of our method with the three well-established learning algorithms Q learning, SARSA learning, and actor-critic learning. Illustrations of their dynamics on two multiagent, multistate environments reveal a wide range of different dynamical regimes, such as convergence to fixed points, limit cycles, and even deterministic chaos.

  • 2. Barfuss, Wolfram
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Lade, Steven J.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Kurths, Jürgen
    When optimization for governing human-environment tipping elements is neither sustainable nor safe2018In: Nature Communications, ISSN 2041-1723, E-ISSN 2041-1723, Vol. 9, article id 2354Article in journal (Refereed)
    Abstract [en]

    Optimizing economic welfare in environmental governance has been criticized for delivering short-term gains at the expense of long-term environmental degradation. Different from economic optimization, the concepts of sustainability and the more recent safe operating space have been used to derive policies in environmental governance. However, a formal comparison between these three policy paradigms is still missing, leaving policy makers uncertain which paradigm to apply. Here, we develop a better understanding of their interrelationships, using a stylized model of human-environment tipping elements. We find that no paradigm guarantees fulfilling requirements imposed by another paradigm and derive simple heuristics for the conditions under which these trade-offs occur. We show that the absence of such a master paradigm is of special relevance for governing real-world tipping systems such as climate, fisheries, and farming, which may reside in a parameter regime where economic optimization is neither sustainable nor safe.

  • 3. Barfuss, Wolfram
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Wiedermann, Marc
    Lucht, Wolfgang
    Sustainable use of renewable resources in a stylized social-ecological network model under heterogeneous resource distribution2017In: Earth System Dynamics, ISSN 2190-4979, E-ISSN 2190-4987, Vol. 8, no 2, p. 255-264Article in journal (Refereed)
    Abstract [en]

    Human societies depend on the resources ecosystems provide. Particularly since the last century, human activities have transformed the relationship between nature and society at a global scale. We study this coevolutionary relationship by utilizing a stylized model of private resource use and social learning on an adaptive network. The latter process is based on two social key dynamics beyond economic paradigms: boundedly rational imitation of resource use strategies and homophily in the formation of social network ties. The private and logistically growing resources are harvested with either a sustainable (small) or non-sustainable (large) effort. We show that these social processes can have a profound influence on the environmental state, such as determining whether the private renewable resources collapse from overuse or not. Additionally, we demonstrate that heterogeneously distributed regional resource capacities shift the critical social parameters where this resource extraction system collapses. We make these points to argue that, in more advanced coevolutionary models of the planetary social-ecological system, such socio-cultural phenomena as well as regional resource heterogeneities should receive attention in addition to the processes represented in established Earth system and integrated assessment models.

  • 4.
    Donges, Jonathan
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany; Humboldt University of Berlin, Germany.
    Donner, Reik V.
    Kurths, Juergen
    Testing time series irreversibility using complex network methods2013In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 102, no 1, p. 10004-Article in journal (Refereed)
    Abstract [en]

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

  • 5.
    Donges, Jonathan F.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany,.
    Barfuss, Wolfram
    From Math to Metaphors and Back Again Social-Ecological Resilience from a Multi-Agent-Environment Perspective2017In: GAIA, ISSN 0940-5550, Vol. 26, p. 182-190Article in journal (Refereed)
    Abstract [en]

    Social-ecological resilience underlies popular sustainability concepts that have been influential in formulating the United Nations Sustainable Development Goals (SDGs), such as the Planetary Boundaries and Doughnut Economics. Scientific investigation of these concepts is supported by mathematical models of planetary biophysical and societal dynamics, both of which call for operational measures of resilience. However, current quantitative descriptions tend to be restricted to the foundational form of the concept: persistence resilience. We propose a classification of modern notions of social-ecological resilience from a multi-agent-environment perspective. This aims at operationalization in a complex systems framework, including the persistence, adaptation and transformation aspects of resilience, normativity related to desirable system function, first-vs. second-order and specific vs. general resilience. For example, we discuss the use of the Topology of Sustainable Management Framework. Developing the mathematics of resilience along these lines would not only make social-ecological resilience more applicable to data and models, but could also conceptually advance resilience thinking.

  • 6.
    Donges, Jonathan F.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Donner, R. V.
    Marwan, N.
    Breitenbach, S. F. M.
    Rehfeld, K.
    Kurths, J.
    Non-linear regime shifts in Holocene Asian monsoon variability: potential impacts on cultural change and migratory patterns2015In: Climate of the Past, ISSN 1814-9324, E-ISSN 1814-9332, Vol. 11, no 5, p. 709-741Article in journal (Refereed)
    Abstract [en]

    The Asian monsoon system is an important tipping element in Earth's climate with a large impact on human societies in the past and present. In light of the potentially severe impacts of present and future anthropogenic climate change on Asian hydrology, it is vital to understand the forcing mechanisms of past climatic regime shifts in the Asian monsoon domain. Here we use novel recurrence network analysis techniques for detecting episodes with pronounced non-linear changes in Holocene Asian monsoon dynamics recorded in speleothems from caves distributed throughout the major branches of the Asian monsoon system. A newly developed multi-proxy methodology explicitly considers dating uncertainties with the COPRA (COnstructing Proxy Records from Age models) approach and allows for detection of continental-scale regime shifts in the complexity of monsoon dynamics. Several epochs are characterised by non-linear regime shifts in Asian monsoon variability, including the periods around 8.5-7.9, 5.7-5.0, 4.1-3.7, and 3.0-2.4 ka BP. The timing of these regime shifts is consistent with known episodes of Holocene rapid climate change (RCC) and high-latitude Bond events. Additionally, we observe a previously rarely reported non-linear regime shift around 7.3 ka BP, a timing that matches the typical 1.0-1.5 ky return intervals of Bond events. A detailed review of previously suggested links between Holocene climatic changes in the Asian monsoon domain and the archaeological record indicates that, in addition to previously considered longer-term changes in mean monsoon intensity and other climatic parameters, regime shifts in monsoon complexity might have played an important role as drivers of migration, pronounced cultural changes, and the collapse of ancient human societies.

  • 7.
    Donges, Jonathan F.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Heitzig, Jobst
    Beronov, Boyan
    Wiedermann, Marc
    Runge, Jakob
    Feng, Qing Yi
    Tupikina, Liubov
    Stolbova, Veronika
    Donner, Reik V.
    Marwan, Norbert
    Dijkstra, Henk A.
    Kurths, Jürgen
    Unified functional network and nonlinear time series analysis for complex systems science: The pyunicorn package2015In: Chaos, ISSN 1054-1500, E-ISSN 1089-7682, Vol. 25, no 11, article id 113101Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Donges, Jonathan F.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Petrova, Irina
    Loew, Alexander
    Marwan, Norbert
    Kurths, Jürgen
    How complex climate networks complement eigen techniques for the statistical analysis of climatological data2015In: Climate Dynamics, ISSN 0930-7575, E-ISSN 1432-0894, Vol. 45, no 9-10, p. 2407-2424Article in journal (Refereed)
    Abstract [en]

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

  • 9.
    Donges, Jonathan F.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Schleussner, C. -F.
    Siegmund, J. F.
    Donner, R. V.
    Event coincidence analysis for quantifying statistical interrelationships between event time series2016In: The European Physical Journal Special Topics, ISSN 1951-6355, E-ISSN 1951-6401, Vol. 225, no 3, p. 471-487Article in journal (Refereed)
    Abstract [en]

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

  • 10. Donner, Reik V.
    et al.
    Stolbova, Veronika
    Balasis, Georgios
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Georgiou, Marina
    Potirakis, Stelios M.
    Kurths, Jürgen
    Temporal organization of magnetospheric fluctuations unveiled by recurrence patterns in the Dst index2018In: Chaos, ISSN 1054-1500, E-ISSN 1089-7682, Vol. 28, no 8, article id 085716Article in journal (Refereed)
    Abstract [en]

    Magnetic storms constitute the most remarkable large-scale phenomena of nonlinear magnetospheric dynamics. Studying the dynamical organization of macroscopic variability in terms of geomagnetic activity index data by means of complexity measures provides a promising approach for identifying the underlying processes and associated time scales. Here, we apply a suite of characteristics from recurrence quantification analysis (RQA) and recurrence network analysis (RNA) in order to unveil some key nonlinear features of the hourly Disturbance storm-time (Dst) index during periods with magnetic storms and such of normal variability. Our results demonstrate that recurrence-based measures can serve as excellent tracers for changes in the dynamical complexity along non-stationary records of geomagnetic activity. In particular, trapping time (characterizing the typical length of laminar phases in the observed dynamics) and recurrence network transitivity (associated with the number of the system's effective dynamical degrees of freedom) allow for a very good discrimination between magnetic storm and quiescence phases. In general, some RQA and RNA characteristics distinguish between storm and non-storm times equally well or even better than other previously considered nonlinear characteristics like Hurst exponent or symbolic dynamics based entropy concepts. Our results point to future potentials of recurrence characteristics for unveiling temporal changes in the dynamical complexity of the magnetosphere.

  • 11.
    Downing, Andrea S.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Bhowmik, Avit
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Collste, David
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Université Clermont Auvergne, France.
    Cornell, Sarah E.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Donges, Jonathan
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Fetzer, Ingo
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Häyhä, Tiina
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. International Institute for Applied Systems Analysis (IIASA), Austria.
    Hinton, Jennifer
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Université Clermont Auvergne, France.
    Lade, Steven
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Mooij, Wolf M.
    Matching scope, purpose and uses of planetary boundaries science2019In: Environmental Research Letters, ISSN 1748-9326, E-ISSN 1748-9326, Vol. 14, no 7, article id 073005Article, review/survey (Refereed)
    Abstract [en]

    Background: The Planetary Boundaries concept (PBc) has emerged as a key global sustainability concept in international sustainable development arenas. Initially presented as an agenda for global sustainability research, it now shows potential for sustainability governance. Weuse the fact that it is widely cited in scientific literature (>3500 citations) and an extensively studied concept to analyse how it has been used and developed since its first publication. Design: From the literature that cites the PBc, we select those articles that have the terms 'planetary boundaries' or 'safe operating space' in either title, abstract or keywords. Weassume that this literature substantively engages with and develops the PBc. Results: Wefind that 6% of the citing literature engages with the concept. Within this fraction of the literature we distinguish commentaries-that discuss the context and challenges to implementing the PBc, articles that develop the core biogeophysical concept and articles that apply the concept by translating to sub-global scales and by adding a human component to it. Applied literature adds to the concept by explicitly including society through perspectives of impacts, needs, aspirations and behaviours. Discussion: Literature applying the concept does not yet include the more complex, diverse, cultural and behavioural facet of humanity that is implied in commentary literature. Wesuggest there is need for a positive framing of sustainability goals-as a Safe Operating Space rather than boundaries. Key scientific challenges include distinguishing generalised from context-specific knowledge, clarifying which processes are generalizable and which are scalable, and explicitly applying complex systems' knowledge in the application and development of the PBc. We envisage that opportunities to address these challenges will arise when more human social dimensions are integrated, as we learn to feed the global sustainability vision with a plurality of bottom-up realisations of sustainability.

  • 12. Feldhoff, J. H.
    et al.
    Donner, R. V.
    Donges, Jonathan
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany; Humboldt University of Berlin, Germany.
    Marwan, N.
    Kurths, J.
    Geometric signature of complex synchronisation scenarios2013In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 102, no 3, p. 30007-Article in journal (Refereed)
    Abstract [en]

    Synchronisation between coupled oscillatory systems is a common phenomenon in many natural as well as technical systems. Varying the coupling strength often leads to qualitative changes in the dynamics exhibiting different types of synchronisation. Here, we study the geometric signatures of coupling along with the onset of generalised synchronisation (GS) between two coupled chaotic oscillators by mapping the systems' individual as well as joint recurrences in phase space to a complex network. For a paradigmatic continuous-time model system, we show that the transitivity properties of the resulting joint recurrence networks display distinct variations associated with changes in the structural similarity between different parts of the considered trajectories. They therefore provide a useful new indicator for the emergence of GS.

  • 13. Feldhoff, Jan H.
    et al.
    Lange, Stefan
    Volkholz, Jan
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Juergen
    Gerstengarbe, Friedrich-Wilhelm
    Complex networks for climate model evaluation with application to statistical versus dynamical modeling of South American climate2015In: Climate Dynamics, ISSN 0930-7575, E-ISSN 1432-0894, Vol. 44, no 06-maj, p. 1567-1581Article in journal (Refereed)
    Abstract [en]

    In this study we introduce two new node-weighted difference measures on complex networks as a tool for climate model evaluation. The approach facilitates the quantification of a model's ability to reproduce the spatial covariability structure of climatological time series. We apply our methodology to compare the performance of a statistical and a dynamical regional climate model simulating the South American climate, as represented by the variables 2 m temperature, precipitation, sea level pressure, and geopotential height field at 500 hPa. For each variable, networks are constructed from the model outputs and evaluated against a reference network, derived from the ERA-Interim reanalysis, which also drives the models. We compare two network characteristics, the (linear) adjacency structure and the (nonlinear) clustering structure, and relate our findings to conventional methods of model evaluation. To set a benchmark, we construct different types of random networks and compare them alongside the climate model networks. Our main findings are: (1) The linear network structure is better reproduced by the statistical model statistical analogue resampling scheme (STARS) in summer and winter for all variables except the geopotential height field, where the dynamical model CCLM prevails. (2) For the nonlinear comparison, the seasonal differences are more pronounced and CCLM performs almost as well as STARS in summer (except for sea level pressure), while STARS performs better in winter for all variables.

  • 14. Fujiwara, Naoya
    et al.
    Kirchen, Kathrin
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Donner, Reik V.
    A perturbation-theoretic approach to Lagrangian flow networks2017In: Chaos, ISSN 1054-1500, E-ISSN 1089-7682, Vol. 27, no 3, article id 035813Article in journal (Refereed)
    Abstract [en]

    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 eigen-system 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.

  • 15. Heck, Vera
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Lucht, Wolfgang
    Collateral transgression of planetary boundaries due to climate engineering by terrestrial carbon dioxide removal2016In: Earth System Dynamics, ISSN 2190-4979, E-ISSN 2190-4987, Vol. 7, no 4, p. 783-796Article in journal (Refereed)
    Abstract [en]

    The planetary boundaries framework provides guidelines for defining thresholds in environmental variables. Their transgression is likely to result in a shift in Earth system functioning away from the relatively stable Holocene state. As the climate system is approaching critical thresholds of atmospheric carbon, several climate engineering methods are discussed, aiming at a reduction of atmospheric carbon concentrations to control the Earth's energy balance. Terrestrial carbon dioxide removal (tCDR) via afforestation or bioenergy production with carbon capture and storage are part of most climate change mitigation scenarios that limit global warming to less than 2 degrees C. We analyse the co-evolutionary interaction of societal interventions via tCDR and the natural dynamics of the Earth's carbon cycle. Applying a conceptual modelling framework, we analyse how the degree of anticipation of the climate problem and the intensity of tCDR efforts with the aim of staying within a safe level of global warming might influence the state of the Earth system with respect to other carbon-related planetary boundaries. Within the scope of our approach, we show that societal management of atmospheric carbon via tCDR can lead to a collateral transgression of the planetary boundary of land system change. Our analysis indicates that the opportunities to remain in a desirable region within carbon-related planetary boundaries only exist for a small range of anticipation levels and depend critically on the underlying emission pathway. While tCDR has the potential to ensure the Earth system's persistence within a carbon-safe operating space under low-emission pathways, it is unlikely to succeed in a business-as-usual scenario.

  • 16. Heitzig, J.
    et al.
    Kittel, T.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Molkenthin, N.
    Topology of sustainable management of dynamical systems with desirable states: from defining planetary boundaries to safe operating spaces in the Earth system2016In: Earth System Dynamics, ISSN 2190-4979, E-ISSN 2190-4987, Vol. 7, no 1, p. 21-50Article in journal (Refereed)
    Abstract [en]

    To keep the Earth system in a desirable region of its state space, such as defined by the recently suggested tolerable environment and development window, guardrails, planetary boundaries, or safe (and just) operating space for humanity, one needs to understand not only the quantitative internal dynamics of the system and the available options for influencing it (management) but also the structure of the system's state space with regard to certain qualitative differences. Important questions are, which state space regions can be reached from which others with or without leaving the desirable region, which regions are in a variety of senses safe to stay in when management options might break away, and which qualitative decision problems may occur as a consequence of this topological structure? In this article, we develop a mathematical theory of the qualitative topology of the state space of a dynamical system with management options and desirable states, as a complement to the existing literature on optimal control which is more focussed on quantitative optimization and is much applied in both the engineering and the integrated assessment literature. We suggest a certain terminology for the various resulting regions of the state space and perform a detailed formal classification of the possible states with respect to the possibility of avoiding or leaving the undesired region. Our results indicate that, before performing some form of quantitative optimization such as of indicators of human well-being for achieving certain sustainable development goals, a sustainable and resilient management of the Earth system may require decisions of a more discrete type that come in the form of several dilemmas, e. g. choosing between eventual safety and uninterrupted desirability, or between uninterrupted safety and larger flexibility. We illustrate the concepts and dilemmas drawing on conceptual models from climate science, ecology, coevolutionary Earth system modelling, economics, and classical mechanics, and discuss their potential relevance for the climate and sustainability debate, in particular suggesting several levels of planetary boundaries of qualitatively increasing safety.

  • 17. Heitzig, Jobst
    et al.
    Barfuss, Wolfram
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    A Thought Experiment on Sustainable Management of the Earth System2018In: Sustainability, ISSN 2071-1050, E-ISSN 2071-1050, Vol. 10, no 6, article id 1947Article in journal (Refereed)
    Abstract [en]

    We introduce and analyze a simple formal thought experiment designed to reflect a qualitative decision dilemma humanity might currently face in view of anthropogenic climate change. In this exercise, each generation can choose between two options, either setting humanity on a pathway to certain high wellbeing after one generation of suffering, or leaving the next generation in the same state as the current one with the same options, but facing a continuous risk of permanent collapse. We analyze this abstract setup regarding the question of what the right choice would be both in a rationality-based framework including optimal control, welfare economics, and game theory, and by means of other approaches based on the notions of responsibility, safe operating spaces, and sustainability paradigms. Across these different approaches, we confirm the intuition that a focus on the long-term future makes the first option more attractive while a focus on equality across generations favors the second. Despite this, we generally find a large diversity and disagreement of assessments both between and within these different approaches, suggesting a strong dependence on the choice of the normative framework used. This implies that policy measures selected to achieve targets such as the United Nations Sustainable Development Goals can depend strongly on the normative framework applied and specific care needs to be taken with regard to the choice of such frameworks.

  • 18. Klamser, Pascal P.
    et al.
    Wiedermann, Marc
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Donner, Reik V.
    Zealotry effects on opinion dynamics in the adaptive voter model2017In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 96, no 5, article id 052315Article in journal (Refereed)
    Abstract [en]

    The adaptive voter model has been widely studied as a conceptual model for opinion formation processes on time-evolving social networks. Past studies on the effect of zealots, i.e., nodes aiming to spread their fixed opinion throughout the system, only considered the voter model on a static network. Here we extend the study of zealotry to the case of an adaptive network topology co-evolving with the state of the nodes and investigate opinion spreading induced by zealots depending on their initial density and connectedness. Numerical simulations reveal that below the fragmentation threshold a low density of zealots is sufficient to spread their opinion to the whole network. Beyond the transition point, zealots must exhibit an increased degree as compared to ordinary nodes for an efficient spreading of their opinion. We verify the numerical findings using a mean-field approximation of the model yielding a low-dimensional set of coupled ordinary differential equations. Our results imply that the spreading of the zealots' opinion in the adaptive voter model is strongly dependent on the link rewiring probability and the average degree of normal nodes in comparison with that of the zealots. In order to avoid a complete dominance of the zealots' opinion, there are two possible strategies for the remaining nodes: adjusting the probability of rewiring and/or the number of connections with other nodes, respectively.

  • 19.
    Lade, Steven J.
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Fetzer, Ingo
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Anderies, John M.
    Beer, Christian
    Stockholm University, Faculty of Science, Department of Environmental Science and Analytical Chemistry.
    Cornell, Sarah E.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Gasser, Thomas
    Norberg, Jon
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Richardson, Katherine
    Rockström, Johan
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Steffen, Will
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Analytically tractable climate-carbon cycle feedbacks under 21st century anthropogenic forcing2018In: Earth System Dynamics, ISSN 2190-4979, E-ISSN 2190-4987, Vol. 9, no 2, p. 507-523Article in journal (Refereed)
    Abstract [en]

    Changes to climate-carbon cycle feedbacks may significantly affect the Earth system's response to greenhouse gas emissions. These feedbacks are usually analysed from numerical output of complex and arguably opaque Earth system models. Here, we construct a stylised global climate-carbon cycle model, test its output against comprehensive Earth system models, and investigate the strengths of its climate-carbon cycle feedbacks analytically. The analytical expressions we obtain aid understanding of carbon cycle feedbacks and the operation of the carbon cycle. Specific results include that different feedback formalisms measure fundamentally the same climate-carbon cycle processes; temperature dependence of the solubility pump, biological pump, and CO2 solubility all contribute approximately equally to the ocean climate-carbon feedback; and concentration-carbon feedbacks may be more sensitive to future climate change than climate-carbon feedbacks. Simple models such as that developed here also provide workbenches for simple but mechanistically based explorations of Earth system processes, such as interactions and feedbacks between the planetary boundaries, that are currently too uncertain to be included in comprehensive Earth system models.

  • 20. Lange, Stefan
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Volkholz, Jan
    Kurths, Juergen
    Local Difference Measures between Complex Networks for Dynamical System Model Evaluation2015In: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 10, no 4, article id e0118088Article in journal (Refereed)
    Abstract [en]

    A faithful modeling of real-world dynamical systems necessitates model evaluation. A recent promising methodological approach to this problem has been based on complex networks, which in turn have proven useful for the characterization of dynamical systems. In this context, we introduce three local network difference measures and demonstrate their capabilities in the field of climate modeling, where these measures facilitate a spatially explicit model evaluation. Building on a recent study by Feldhoff et al. [1] we comparatively analyze statistical and dynamical regional climate simulations of the South American monsoon system. Three types of climate networks representing different aspects of rainfall dynamics are constructed from the modeled precipitation space-time series. Specifically, we define simple graphs based on positive as well as negative rank correlations between rainfall anomaly time series at different locations, and such based on spatial synchronizations of extreme rain events. An evaluation against respective networks built from daily satellite data provided by the Tropical Rainfall Measuring Mission 3B42 V7 reveals far greater differences in model performance between network types for a fixed but arbitrary climate model than between climate models for a fixed but arbitrary network type. We identify two sources of uncertainty in this respect. Firstly, climate variability limits fidelity, particularly in the case of the extreme event network; and secondly, larger geographical link lengths render link misplacements more likely, most notably in the case of the anticorrelation network; both contributions are quantified using suitable ensembles of surrogate networks. Our model evaluation approach is applicable to any multidimensional dynamical system and especially our simple graph difference measures are highly versatile as the graphs to be compared may be constructed in whatever way required. Generalizations to directed as well as edge-and node-weighted graphs are discussed.

  • 21. Mahecha, Miguel D.
    et al.
    Gans, Fabian
    Sippel, Sebastian
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kaminski, Thomas
    Metzger, Stefan
    Migliavacca, Mirco
    Papale, Dario
    Rammig, Anja
    Zscheischler, Jakob
    Detecting impacts of extreme events with ecological in situ monitoring networks2017In: Biogeosciences, ISSN 1726-4170, E-ISSN 1726-4189, Vol. 14, no 18, p. 4255-4277Article in journal (Refereed)
    Abstract [en]

    Extreme hydrometeorological conditions typically impact ecophysiological processes on land. Satellite-based observations of the terrestrial biosphere provide an important reference for detecting and describing the spatiotemporal development of such events. However, in-depth investigations of ecological processes during extreme events require additional in situ observations. The question is whether the density of existing ecological in situ networks is sufficient for analysing the impact of extreme events, and what are expected event detection rates of ecological in situ networks of a given size. To assess these issues, we build a baseline of extreme reductions in the fraction of absorbed photosynthetically active radiation (FAPAR), identified by a new event detection method tailored to identify extremes of regional relevance. We then investigate the event detection success rates of hypothetical networks of varying sizes. Our results show that large extremes can be reliably detected with relatively small networks, but also reveal a linear decay of detection probabilities towards smaller extreme events in log-log space. For instance, networks with approximate to 100 randomly placed sites in Europe yield a >= 90% chance of detecting the eight largest (typically very large) extreme events; but only a >= 50% chance of capturing the 39 largest events. These findings are consistent with probability-theoretic considerations, but the slopes of the decay rates deviate due to temporal autocorrelation and the exact implementation of the extreme event detection algorithm. Using the examples of AmeriFlux and NEON, we then investigate to what degree ecological in situ networks can capture extreme events of a given size. Consistent with our theoretical considerations, we find that today's systematically designed networks (i.e. NEON) reliably detect the largest extremes, but that the extreme event detection rates are not higher than would be achieved by randomly designed networks. Spatio-temporal expansions of ecological in situ monitoring networks should carefully consider the size distribution characteristics of extreme events if the aim is also to monitor the impacts of such events in the terrestrial biosphere.

  • 22. Milkoreit, Manjana
    et al.
    Hodbod, Jennifer
    Baggio, Jacopo
    Benessaiah, Karina
    Calderón-Contreras, Rafael
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Mathias, Jean-Denis
    Rocha, Juan Carlos
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Schoon, Michael
    Werners, Saskia E.
    Defining tipping points for social-ecological systems scholarship-an interdisciplinary literature review2018In: Environmental Research Letters, ISSN 1748-9326, E-ISSN 1748-9326, Vol. 13, no 3, article id 033005Article, review/survey (Refereed)
    Abstract [en]

    The term tipping point has experienced explosive popularity across multiple disciplines over the last decade. Research on social-ecological systems (SES) has contributed to the growth and diversity of the term's use. The diverse uses of the term obscure potential differences between tipping behavior in natural and social systems, and issues of causality across natural and social system components in SES. This paper aims to create the foundation for a discussion within the SES research community about the appropriate use of the term tipping point, especially the relatively novel term 'social tipping point.' We review existing literature on tipping points and similar concepts (e.g. regime shifts, critical transitions) across all spheres of science published between 1960 and 2016 with a special focus on a recent and still small body of work on social tipping points. We combine quantitative and qualitative analyses in a bibliometric approach, rooted in an expert elicitation process. We find that the term tipping point became popular after the year 2000-long after the terms regime shift and critical transition-across all spheres of science. We identify 23 distinct features of tipping point definitions and their prevalence across disciplines, but find no clear taxonomy of discipline-specific definitions. Building on the most frequently used features, we propose definitions for tipping points in general and social tipping points in SES in particular.

  • 23. Molkenthin, Nora
    et al.
    Kutza, Hannes
    Tupikina, Liubov
    Marwan, Norbert
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Feudel, Ulrike
    Kurths, Jürgen
    Donner, Reik V.
    Edge anisotropy and the geometric perspective on flow networks2017In: Chaos, ISSN 1054-1500, E-ISSN 1089-7682, Vol. 27, no 3, article id 035802Article in journal (Refereed)
    Abstract [en]

    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 the 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 spatiotemporal 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.

  • 24. Müller-Hansen, Finn
    et al.
    Cardoso, Manoel F.
    Dalla-Nora, Eloi L.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Heitzig, Jobst
    Kurths, Jürgen
    Thonicke, Kirsten
    A matrix clustering method to explore patterns of land-cover transitions in satellite-derived maps of the Brazilian Amazon2017In: Nonlinear processes in geophysics, ISSN 1023-5809, E-ISSN 1607-7946, Vol. 24, no 1, p. 113-123Article in journal (Refereed)
    Abstract [en]

    Changes in land-use systems in tropical regions, including deforestation, are a key challenge for global sustainability because of their huge impacts on green-house gas emissions, local climate and biodiversity. However, the dynamics of land-use and land-cover change in regions of frontier expansion such as the Brazilian Amazon are not yet well understood because of the complex interplay of ecological and socioeconomic drivers. In this paper, we combine Markov chain analysis and complex network methods to identify regimes of land-cover dynamics from land-cover maps (TerraClass) derived from high-resolution (30 m) satellite imagery. We estimate regional transition probabilities between different land-cover types and use clustering analysis and community detection algorithms on similarity networks to explore patterns of dominant land- cover transitions. We find that land- cover transition probabilities in the Brazilian Amazon are heterogeneous in space, and adjacent subregions tend to be assigned to the same clusters. When focusing on transitions from single land- cover types, we uncover patterns that reflect major regional differences in land-cover dynamics. Our method is able to summarize regional patterns and thus complements studies performed at the local scale.

  • 25. Müller-Hansen, Finn
    et al.
    Heitzig, Jobst
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research (PIK), Germany.
    Cardoso, Manoel F.
    Dalla-Nora, Eloi L.
    Andrade, Pedro
    Kurths, Jürgen
    Thonicke, Kirsten
    Can Intensification of Cattle Ranching Reduce Deforestation in the Amazon? Insights From an Agent-based Social-Ecological Model2019In: Ecological Economics, ISSN 0921-8009, E-ISSN 1873-6106, Vol. 159, p. 198-211Article in journal (Refereed)
    Abstract [en]

    Deforestation in the Amazon with its vast consequences for the ecosystem and climate is largely related to subsequent land use for cattle ranching. In addition to conservation policies, proposals to reduce deforestation include measures to intensify cattle ranching. However, the effects of land-use intensification on deforestation are debated in the literature. This paper introduces the abacra model, a stylized agent-based model to study the interplay of deforestation and the intensification of cattle ranching in the Brazilian Amazon. The model combines social learning and ecological processes with market dynamics. In the model, agents adopt either an extensive or semi-intensive strategy of cattle ranching based on the success of their neighbors. They earn their income by selling cattle on a stylized market. We present a comprehensive analysis of the model with statistical methods and find that it produces highly non-linear transient outcomes in dependence on key parameters like the rate of social interaction and elasticity of the cattle price. We show that under many environmental and economic conditions, intensification does not reduce deforestation rates and sometimes even has a detrimental effect on deforestation. Anti-deforestation policies incentivizing fast intensification can only lower deforestation rates under conditions in which the local cattle market saturates.

  • 26. Müller-Hansen, Finn
    et al.
    Schlüter, Maja
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Mäs, Michael
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research (PIK), Germany.
    Kolb, Jakob J.
    Thonicke, Kirsten
    Heitzig, Jobst
    Towards representing human behavior and decision making in Earth system models - an overview of techniques and approaches2017In: Earth System Dynamics, ISSN 2190-4979, E-ISSN 2190-4987, Vol. 8, no 4, p. 977-1007Article in journal (Refereed)
    Abstract [en]

    Today, humans have a critical impact on the Earth system and vice versa, which can generate complex feedback processes between social and ecological dynamics. Integrating human behavior into formal Earth system models (ESMs), however, requires crucial modeling assumptions about actors and their goals, behavioral options, and decision rules, as well as modeling decisions regarding human social interactions and the aggregation of individuals' behavior. Here, we review existing modeling approaches and techniques from various disciplines and schools of thought dealing with human behavior at different levels of decision making. We demonstrate modelers' often vast degrees of freedom but also seek to make modelers aware of the often crucial consequences of seemingly innocent modeling assumptions. After discussing which socioeconomic units are potentially important for ESMs, we compare models of individual decision making that correspond to alternative behavioral theories and that make diverse modeling assumptions about individuals' preferences, beliefs, decision rules, and foresight. We review approaches to model social interaction, covering game theoretic frameworks, models of social influence, and network models. Finally, we discuss approaches to studying how the behavior of individuals, groups, and organizations can aggregate to complex collective phenomena, discussing agent-based, statistical, and representative-agent modeling and economic macro-dynamics. We illustrate the main ingredients of modeling techniques with examples from land-use dynamics as one of the main drivers of environmental change bridging local to global scales.

  • 27. Nocke, T.
    et al.
    Buschmann, S.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Marwan, N.
    Schulz, H. -J.
    Tominski, C.
    Review: visual analytics of climate networks2015In: Nonlinear processes in geophysics, ISSN 1023-5809, E-ISSN 1607-7946, Vol. 22, no 5, p. 545-570Article, review/survey (Refereed)
    Abstract [en]

    Network analysis has become an important approach in studying complex spatiotemporal behaviour within geophysical observation and simulation data. This new field produces increasing numbers of large geo-referenced networks to be analysed. Particular focus lies currently on the network analysis of the complex statistical interrelationship structure within climatological fields. The standard procedure for such network analyses is the extraction of network measures in combination with static standard visualisation methods. Existing interactive visualisation methods and tools for geo-referenced network exploration are often either not known to the analyst or their potential is not fully exploited. To fill this gap, we illustrate how interactive visual analytics methods in combination with geovisualisation can be tailored for visual climate network investigation. Therefore, the paper provides a problem analysis relating the multiple visualisation challenges to a survey undertaken with network analysts from the research fields of climate and complex systems science. Then, as an overview for the interested practitioner, we review the state-of-the-art in climate network visualisation and provide an overview of existing tools. As a further contribution, we introduce the visual network analytics tools CGV and GTX, providing tailored solutions for climate network analysis, including alternative geographic projections, edge bundling, and 3-D network support. Using these tools, the paper illustrates the application potentials of visual analytics for climate networks based on several use cases including examples from global, regional, and multi-layered climate networks.

  • 28. Radebach, A.
    et al.
    Donner, R. V.
    Runge, J.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. University of Berlin, Germany.
    Kurths, J.
    Disentangling different types of El Nino episodes by evolving climate network analysis2013In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 88, no 5, p. 052807-Article in journal (Refereed)
    Abstract [en]

    Complex network theory provides a powerful toolbox for studying the structure of statistical interrelationships between multiple time series in various scientific disciplines. In this work, we apply the recently proposed climate network approach for characterizing the evolving correlation structure of the Earth's climate system based on reanalysis data for surface air temperatures. We provide a detailed study of the temporal variability of several global climate network characteristics. Based on a simple conceptual view of red climate networks (i.e., networks with a comparably low number of edges), we give a thorough interpretation of our evolving climate network characteristics, which allows a functional discrimination between recently recognized different types of El Nino episodes. Our analysis provides deep insights into the Earth's climate system, particularly its global response to strong volcanic eruptions and large-scale impacts of different phases of the El Nino Southern Oscillation.

  • 29. Rammig, A.
    et al.
    Wiedermann, M.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Babst, F.
    von Bloh, W.
    Frank, D.
    Thonicke, K.
    Mahecha, M. D.
    Coincidences of climate extremes and anomalous vegetation responses: comparing tree ring patterns to simulated productivity2015In: Biogeosciences, ISSN 1726-4170, E-ISSN 1726-4189, Vol. 12, no 2, p. 373-385Article in journal (Refereed)
    Abstract [en]

    Climate extremes can trigger exceptional responses in terrestrial ecosystems, for instance by altering growth or mortality rates. Such effects are often manifested in reductions in net primary productivity (NPP). Investigating a Europe-wide network of annual radial tree growth records confirms this pattern: we find that 28% of tree ring width (TRW) indices are below two standard deviations in years in which extremely low precipitation, high temperatures or the combination of both noticeably affect tree growth. Based on these findings, we investigate possibilities for detecting climate-driven patterns in long-term TRW data to evaluate state-of-the-art dynamic vegetation models such as the Lund-Potsdam-Jena dynamic global vegetation model for managed land (LPJmL). The major problem in this context is that LPJmL simulates NPP but not explicitly the radial tree growth, and we need to develop a generic method to allow for a comparison between simulated and observed response patterns. We propose an analysis scheme that quantifies the coincidence rate of climate extremes with some biotic responses (here TRW or simulated NPP). We find a relative reduction of 34% in simulated NPP during precipitation, temperature and combined extremes. This reduction is comparable to the TRW response patterns, but the model responds much more sensitively to drought stress. We identify 10 extreme years during the 20th century during which both model and measurements indicate high coincidence rates across Europe. However, we detect substantial regional differences in simulated and observed responses to climatic extreme events. One explanation for this discrepancy could be the tendency of tree ring data to originate from climatically stressed sites. The difference between model and observed data is amplified by the fact that dynamic vegetation models are designed to simulate mean ecosystem responses on landscape or regional scales. We find that both simulation results and measurements display carry-over effects from climate anomalies during the previous year. We conclude that radial tree growth chronologies provide a suitable basis for generic model benchmarks. The broad application of coincidence analysis in generic model benchmarks along with an increased availability of representative long-term measurements and improved process-based models will refine projections of the long-term carbon balance in terrestrial ecosystems.

  • 30. Runge, Jakob
    et al.
    Petoukhov, Vladimir
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Hlinka, Jaroslav
    Jajcay, Nikola
    Vejmelka, Martin
    Hartman, David
    Marwan, Norbert
    Palus, Milan
    Kurths, Juergen
    Identifying causal gateways and mediators in complex spatio-temporal systems2015In: Nature Communications, ISSN 2041-1723, E-ISSN 2041-1723, Vol. 6, article id 8502Article in journal (Refereed)
    Abstract [en]

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

  • 31. Schleussner, C. -F.
    et al.
    Divine, D. V.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Miettinen, A.
    Donner, R. V.
    Indications for a North Atlantic ocean circulation regime shift at the onset of the Little Ice Age2015In: Climate Dynamics, ISSN 0930-7575, E-ISSN 1432-0894, Vol. 45, no 11-12, p. 3623-3633Article in journal (Refereed)
    Abstract [en]

    A prominent characteristic of the reconstructed Northern Hemisphere temperature signal over the last millennium is the transition from the Medieval Climate Anomaly to the Little Ice Age (LIA). Here we report indications for a non-linear regime shift in the North Atlantic ocean circulation at the onset of the LIA. Specifically, we apply a novel statistical test based on horizontal visibility graphs to two ocean sediment August sea-surface temperature records from the Norwegian Sea and the central subpolar basin and find robust indications of time-irreversibility in both records during the LIA onset. Despite a basin-wide cooling trend, we report an anomalous warming in the central subpolar basin during the LIA that is reproduced in ensemble simulations with the climate model of intermediate complexity CLIMBER-3 as a result of a non-linear regime shift in the subpolar North Atlantic ocean circulation. The identified volcanically triggered non-linear transition in the model simulations provides a plausible explanation for the signatures of time-irreversibility found in the ocean sediment records. Our findings indicate a potential multi-stability of the North Atlantic ocean circulation and its importance for regional climate change on centennial time scales.

  • 32. Schleussner, Carl-Friedrich
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Donner, Reik V.
    Schellnhuber, Hans Joachim
    Armed-conflict risks enhanced by climate-related disasters in ethnically fractionalized countries2016In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 113, no 33, p. 9216-9221Article in journal (Refereed)
    Abstract [en]

    Social and political tensions keep on fueling armed conflicts around the world. Although each conflict is the result of an individual context-specific mixture of interconnected factors, ethnicity appears to play a prominent and almost ubiquitous role in many of them. This overall state of affairs is likely to be exacerbated by anthropogenic climate change and in particular climate-related natural disasters. Ethnic divides might serve as predetermined conflict lines in case of rapidly emerging societal tensions arising from disruptive events like natural disasters. Here, we hypothesize that climate-related disaster occurrence enhances armed-conflict outbreak risk in ethnically fractionalized countries. Using event coincidence analysis, we test this hypothesis based on data on armed-conflict outbreaks and climate-related natural disasters for the period 1980-2010. Globally, we find a coincidence rate of 9% regarding armed-conflict outbreak and disaster occurrence such as heat waves or droughts. Our analysis also reveals that, during the period in question, about 23% of conflict outbreaks in ethnically highly fractionalized countries robustly coincide with climatic calamities. Although we do not report evidence that climate-related disasters act as direct triggers of armed conflicts, the disruptive nature of these events seems to play out in ethnically fractionalized societies in a particularly tragic way. This observation has important implications for future security policies as several of the world's most conflict-prone regions, including North and Central Africa as well as Central Asia, are both exceptionally vulnerable to anthropogenic climate change and characterized by deep ethnic divides.

  • 33. Schleussner, Carl-Friedrich
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Engemann, Denis A.
    Levermann, Anders
    Clustered marginalization of minorities during social transitions induced by co-evolution of behaviour and network structure2016In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 6, article id 30790Article in journal (Refereed)
    Abstract [en]

    Large-scale transitions in societies are associated with both individual behavioural change and restructuring of the social network. These two factors have often been considered independently, yet recent advances in social network research challenge this view. Here we show that common features of societal marginalization and clustering emerge naturally during transitions in a co-evolutionary adaptive network model. This is achieved by explicitly considering the interplay between individual interaction and a dynamic network structure in behavioural selection. We exemplify this mechanism by simulating how smoking behaviour and the network structure get reconfigured by changing social norms. Our results are consistent with empirical findings: The prevalence of smoking was reduced, remaining smokers were preferentially connected among each other and formed increasingly marginalized clusters. We propose that self-amplifying feedbacks between individual behaviour and dynamic restructuring of the network are main drivers of the transition. This generative mechanism for co-evolution of individual behaviour and social network structure may apply to a wide range of examples beyond smoking.

  • 34. Siegmund, Jonatan F.
    et al.
    Wiedermann, Marc
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Donner, Reik V.
    Impact of temperature and precipitation extremes on the flowering dates of four German wildlife shrub species2016In: Biogeosciences, ISSN 1726-4170, E-ISSN 1726-4189, Vol. 13, no 19, p. 5541-5555Article in journal (Refereed)
    Abstract [en]

    Ongoing climate change is known to cause an increase in the frequency and amplitude of local temperature and precipitation extremes in many regions of the Earth. While gradual changes in the climatological conditions have already been shown to strongly influence plant flowering dates, the question arises if and how extremes specifically impact the timing of this important phenological phase. Studying this question calls for the application of statistical methods that are tailored to the specific properties of event time series. Here, we employ event coincidence analysis, a novel statistical tool that allows assessing whether or not two types of events exhibit similar sequences of occurrences in order to systematically quantify simultaneities between meteorological extremes and the timing of the flowering of four shrub species across Germany. Our study confirms previous findings of experimental studies by highlighting the impact of early spring temperatures on the flowering of the investigated plants. However, previous studies solely based on correlation analysis do not allow deriving explicit estimates of the strength of such interdependencies without further assumptions, a gap that is closed by our analysis. In addition to direct impacts of extremely warm and cold spring temperatures, our analysis reveals statistically significant indications of an influence of temperature extremes in the autumn preceding the flowering.

  • 35.
    Steffen, Will
    et al.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Rockström, Johan
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Richardson, Katherine
    Lenton, Timothy M.
    Folke, Carl
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Royal Swedish Academy of Science, Sweden.
    Liverman, Diana
    Summerhayes, Colin P.
    Barnosky, Anthony D.
    Cornell, Sarah E.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Crucifix, Michel
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Fetzer, Ingo
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Lade, Steven J.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Scheffer, Marten
    Winkelmann, Ricarda
    Schellnhuber, Hans Joachim
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. The Australian National University, Australia.
    Trajectories of the Earth System in the Anthropocene2018In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 115, no 33, p. 8252-8259Article in journal (Refereed)
    Abstract [en]

    We explore the risk that self-reinforcing feedbacks could push the Earth System toward a planetary threshold that, if crossed, could prevent stabilization of the climate at intermediate temperature rises and cause continued warming on a Hothouse Earth pathway even as human emissions are reduced. Crossing the threshold would lead to a much higher global average temperature than any interglacial in the past 1.2 million years and to sea levels significantly higher than at any time in the Holocene. We examine the evidence that such a threshold might exist and where it might be. If the threshold is crossed, the resulting trajectory would likely cause serious disruptions to ecosystems, society, and economies. Collective human action is required to steer the Earth System away from a potential threshold and stabilize it in a habitable interglacial-like state. Such action entails stewardship of the entire Earth System-biosphere, climate, and societies-and could include decarbonization of the global economy, enhancement of biosphere carbon sinks, behavioral changes, technological innovations, new governance arrangements, and transformed social values.

  • 36. Subramaniyam, N. P.
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Hyttinen, J.
    Signatures of chaotic and stochastic dynamics uncovered with epsilon-recurrence networks2015In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 471, no 2183, article id 20150349Article in journal (Refereed)
    Abstract [en]

    An old and important problem in the field of nonlinear time-series analysis entails the distinction between chaotic and stochastic dynamics. Recently, e-recurrence networks have been proposed as a tool to analyse the structural properties of a time series. In this paper, we propose the applicability of local and global e-recurrence network measures to distinguish between chaotic and stochastic dynamics using paradigmatic model systems such as the Lorenz system, and the chaotic and hyper-chaotic Rossler system. We also demonstrate the effect of increasing levels of noise on these network measures and provide a real-world application of analysing electroencephalographic data comprising epileptic seizures. Our results show that both local and global e-recurrence network measures are sensitive to the presence of unstable periodic orbits and other structural features associated with chaotic dynamics that are otherwise absent in stochastic dynamics. These network measures are still robust at high noise levels and short data lengths. Furthermore, e-recurrence network analysis of the real-world epileptic data revealed the capability of these network measures in capturing dynamical transitions using short window sizes. e-recurrence network analysis is a powerful method in uncovering the signatures of chaotic and stochastic dynamics based on the geometrical properties of time series.

  • 37. Sun, A. Y.
    et al.
    Chen, J.
    Donges, Jonathan
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Global terrestrial water storage connectivity revealed using complex climate network analyses2015In: Nonlinear processes in geophysics, ISSN 1023-5809, E-ISSN 1607-7946, Vol. 22, no 4, p. 433-446Article in journal (Refereed)
    Abstract [en]

    Terrestrial water storage (TWS) exerts a key control in global water, energy, and biogeochemical cycles. Although certain causal relationship exists between precipitation and TWS, the latter quantity also reflects impacts of anthropogenic activities. Thus, quantification of the spatial patterns of TWS will not only help to understand feed-backs between climate dynamics and the hydrologic cycle, but also provide new insights and model calibration constraints for improving the current land surface models. This work is the first attempt to quantify the spatial connectivity of TWS using the complex network theory, which has received broad attention in the climate modeling community in recent years. Complex networks of TWS anomalies are built using two global TWS data sets, a remote sensing product that is obtained from the Gravity Recovery and Climate Experiment (GRACE) satellite mission, and a model-generated data set from the global land data assimilation system's NOAH model (GLDAS-NOAH). Both data sets have 1 degrees x 1 degrees grid resolutions and cover most global land areas except for permafrost regions. TWS networks are built by first quantifying pairwise correlation among all valid TWS anomaly time series, and then applying a cutoff threshold derived from the edge-density function to retain only the most important features in the network. Basinwise network connectivity maps are used to illuminate connectivity of individual river basins with other regions. The constructed network degree centrality maps show the TWS anomaly hotspots around the globe and the patterns are consistent with recent GRACE studies. Parallel analyses of networks constructed using the two data sets reveal that the GLDAS-NOAH model captures many of the spatial patterns shown by GRACE, although significant discrepancies exist in some regions. Thus, our results provide further measures for constraining the current land surface models, especially in data sparse regions.

  • 38. van Kan, Adrian
    et al.
    Jegminat, Jannes
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Juergen
    Constrained basin stability for studying transient phenomena in dynamical systems2016In: Physical Review E, ISSN 2470-0045, Vol. 93, no 4Article in journal (Refereed)
    Abstract [en]

    Transient dynamics are of large interest in many areas of science. Here, a generalization of basin stability (BS) is presented: constrained basin stability (CBS) that is sensitive to various different types of transients arising from finite size perturbations. CBS is applied to the paradigmatic Lorenz system for uncovering nonlinear precursory phenomena of a boundary crisis bifurcation. Further, CBS is used in a model of the Earth's carbon cycle as a return time-dependent stability measure of the system's global attractor. Both case studies illustrate how CBS's sensitivity to transients complements BS in its function as an early warning signal and as a stability measure. CBS is broadly applicable in systems where transients matter, from physics and engineering to sustainability science. Thus CBS complements stability analysis with BS as well as classical linear stability analysis and will be a useful tool for many applications.

  • 39. Wiedermann, M.
    et al.
    Donges, Jonathan
    Stockholm University, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research.
    Heitzig, J.
    Kurths, J.
    Node-weighted interacting network measures improve the representation of real-world complex systems2013In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 102, no 2, p. 28007-Article in journal (Refereed)
    Abstract [en]

    Many real-world complex systems are adequately represented by networks of interacting or interdependent networks. Additionally, it is often reasonable to take into account node weights such as surface area in climate networks, volume in brain networks, or economic capacity in trade networks to reflect the varying size or importance of subsystems. Combining both ideas, we derive a novel class of statistical measures for analysing the structure of networks of interacting networks with heterogeneous node weights. Using a prototypical spatial network model, we show that the newly introduced node-weighted interacting network measures provide an improved representation of the underlying system's properties as compared to their unweighted analogues. We apply our method to study the complex network structure of cross-boundary trade between European Union (EU) and non-EU countries finding that it provides relevant information on trade balance and economic robustness.

  • 40. Wiedermann, Marc
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Handorf, Dörthe
    Kurths, Jürgen
    Donner, Reik V.
    Hierarchical structures in Northern Hemispheric extratropical winter ocean-atmosphere interactions2017In: International Journal of Climatology, ISSN 0899-8418, E-ISSN 1097-0088, Vol. 37, no 10, p. 3821-3836Article in journal (Refereed)
    Abstract [en]

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

  • 41. Wiedermann, Marc
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre.
    Heitzig, Jobst
    Lucht, Wolfgang
    Kurths, Juergen
    Macroscopic description of complex adaptive networks coevolving with dynamic node states2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, no 5, article id 052801Article in journal (Refereed)
    Abstract [en]

    In many real-world complex systems, the time evolution of the network's structure and the dynamic state of its nodes are closely entangled. Here we study opinion formation and imitation on an adaptive complex network which is dependent on the individual dynamic state of each node and vice versa to model the coevolution of renewable resources with the dynamics of harvesting agents on a social network. The adaptive voter model is coupled to a set of identical logistic growth models and we mainly find that, in such systems, the rate of interactions between nodes as well as the adaptive rewiring probability are crucial parameters for controlling the sustainability of the system's equilibrium state. We derive a macroscopic description of the system in terms of ordinary differential equations which provides a general framework to model and quantify the influence of single node dynamics on the macroscopic state of the network. The thus obtained framework is applicable to many fields of study, such as epidemic spreading, opinion formation, or socioecological modeling.

  • 42. Wiedermann, Marc
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Jürgen
    Donner, Reik V.
    Mapping and discrimination of networks in the complexity-entropy plane2017In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 96, no 4, article id 042304Article in journal (Refereed)
    Abstract [en]

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

  • 43. Wiedermann, Marc
    et al.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Jürgen
    Donner, Reik V.
    Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes2016In: Physical Review E, ISSN 2470-0045, Vol. 93, no 4, article id 042308Article in journal (Refereed)
    Abstract [en]

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

  • 44. Wiedermann, Marc
    et al.
    Radebach, Alexander
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Kurths, Juergen
    Donner, Reik V.
    A climate network-based index to discriminate different types of El Nino and La Nina2016In: Geophysical Research Letters, ISSN 0094-8276, E-ISSN 1944-8007, Vol. 43, no 13, p. 7176-7185Article in journal (Refereed)
    Abstract [en]

    El Nino exhibits distinct Eastern Pacific (EP) and Central Pacific (CP) types which are commonly, but not always consistently, distinguished from each other by different signatures in equatorial climate variability. Here we propose an index based on evolving climate networks to objectively discriminate between both flavors by utilizing a scalar-valued measure that quantifies spatial localization and dispersion in global teleconnections of surface air temperature. Our index displays a sharp peak (high localization) during EP events, whereas during CP events (larger dispersion) it remains close to the values observed during normal periods. In contrast to previous classification schemes, our approach specifically accounts for El Nino's global impacts. We confirm recent El Nino classifications for theyears 1951 to 2014 and assign types to those cases where former works yielded ambiguous results. Ultimately, we demonstrate that our index provides a similar discrimination of La Nina episodes into two distinct types.

  • 45. Zemp, D. C.
    et al.
    Schleussner, C. F.
    Barbosa, H. M. J.
    van der Ent, R. J.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research (PIK), Germany.
    Heinke, J.
    Sampaio, G.
    Rammig, A.
    On the importance of cascading moisture recycling in South America2014In: Atmospheric Chemistry And Physics, ISSN 1680-7316, E-ISSN 1680-7324, Vol. 14, no 23, p. 13337-13359Article in journal (Refereed)
    Abstract [en]

    Continental moisture recycling is a crucial process of the South American climate system. In particular, evapotranspiration from the Amazon basin contributes substantially to precipitation regionally as well as over other remote regions such as the La Plata basin. Here we present an in-depth analysis of South American moisture recycling mechanisms. In particular, we quantify the importance of cascading moisture recycling (CMR), which describes moisture transport between two locations on the continent that involves re-evaporation cycles along the way. Using an Eulerian atmospheric moisture tracking model forced by a combination of several historical climate data sets, we were able to construct a complex network of moisture recycling for South America. Our results show that CMR contributes about 9-10% to the total precipitation over South America and 17-18% over the La Plata basin. CMR increases the fraction of total precipitation over the La Plata basin that originates from the Amazon basin from 18-23 to 24-29% during the wet season. We also show that the south-western part of the Amazon basin is not only a direct source of rainfall over the La Plata basin, but also a key intermediary region that distributes moisture originating from the entire Amazon basin towards the La Plata basin during the wet season. Our results suggest that land use change in this region might have a stronger impact on downwind rainfall than previously thought. Using complex network analysis techniques, we find the eastern side of the sub-tropical Andes to be a key region where CMR pathways are channeled. This study offers a better understanding of the interactions between the vegetation and the atmosphere on the water cycle, which is needed in a context of land use and climate change in South America.

  • 46. Zemp, D. C.
    et al.
    Wiedermann, M.
    Kurths, J.
    Rammig, A.
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research, Germany.
    Node-weighted measures for complex networks with directed and weighted edges for studying continental moisture recycling2014In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 107, no 5, p. 58005-Article in journal (Refereed)
    Abstract [en]

    In many real-world networks nodes represent agents or objects of different sizes or importance. However, the size of the nodes is rarely taken into account in network analysis, possibly inducing bias in network measures and confusion in their interpretation. Recently, a new axiomatic scheme of node-weighted network measures has been suggested for networks with undirected and unweighted edges. However, many real-world systems are best represented by complex networks which have directed and/or weighted edges. Here, we extend this approach and suggest new versions of the degree and the clustering coefficient associated to network motifs for networks with directed and/or weighted edges and weighted nodes. We apply these measures to a spatially embedded network model and a real-world moisture recycling network. We show that these measures improve the representation of the underlying systems' structure and are of general use for studying any type of complex network.

  • 47. Zou, Yong
    et al.
    Donner, Reik V.
    Marwan, Norbert
    Donges, Jonathan F.
    Stockholm University, Faculty of Science, Stockholm Resilience Centre. Potsdam Institute for Climate Impact Research (PIK), Germany.
    Kurths, Jürgen
    Complex network approaches to nonlinear time series analysis2019In: Physics reports, ISSN 0370-1573, E-ISSN 1873-6270, Vol. 787, p. 1-97Article, review/survey (Refereed)
    Abstract [en]

    In the last decade, there has been a growing body of literature addressing the utilization of complex network methods for the characterization of dynamical systems based on time series. While both nonlinear time series analysis and complex network theory are widely considered to be established fields of complex systems sciences with strong links to nonlinear dynamics and statistical physics, the thorough combination of both approaches has become an active field of nonlinear time series analysis, which has allowed addressing fundamental questions regarding the structural organization of nonlinear dynamics as well as the successful treatment of a variety of applications from a broad range of disciplines. In this report, we provide an in-depth review of existing approaches of time series networks, covering their methodological foundations, interpretation and practical considerations with an emphasis on recent developments. After a brief outline of the state-of-the-art of nonlinear time series analysis and the theory of complex networks, we focus on three main network approaches, namely, phase space based recurrence networks, visibility graphs and Markov chain based transition networks, all of which have made their way from abstract concepts to widely used methodologies. These three concepts, as well as several variants thereof will be discussed in great detail regarding their specific properties, potentials and limitations. More importantly, we emphasize which fundamental new insights complex network approaches bring into the field of nonlinear time series analysis. In addition, we summarize examples from the wide range of recent applications of these methods, covering rather diverse fields like climatology, fluid dynamics, neurophysiology, engineering and economics, and demonstrating the great potentials of time series networks for tackling real-world contemporary scientific problems. The overall aim of this report is to provide the readers with the knowledge how the complex network approaches can be applied to their own field of real-world time series analysis.

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