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Schunck F, Wiedermann M, Heitzig J, Donges JF. A Dynamic Network Model of Societal Complexity and Resilience Inspired by Tainter's Theory of Collapse. Entropy (Basel) 2024; 26:98. [PMID: 38392354 DOI: 10.3390/e26020098] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2023] [Revised: 01/11/2024] [Accepted: 01/17/2024] [Indexed: 02/24/2024]
Abstract
In recent years, several global events have severely disrupted economies and social structures, undermining confidence in the resilience of modern societies. Examples include the COVID-19 pandemic, which brought unprecedented health challenges and economic disruptions, and the emergence of geopolitical tensions and conflicts that have further strained international relations and economic stability. While empirical evidence on the dynamics and drivers of past societal collapse is mounting, a process-based understanding of these dynamics is still in its infancy. Here, we aim to identify and illustrate the underlying drivers of such societal instability or even collapse. The inspiration for this work is Joseph Tainter's theory of the "collapse of complex societies", which postulates that the complexity of societies increases as they solve problems, leading to diminishing returns on complexity investments and ultimately to collapse. In this work, we abstract this theory into a low-dimensional and stylized model of two classes of networked agents, hereafter referred to as "laborers" and "administrators". We numerically model the dynamics of societal complexity, measured as the fraction of "administrators", which was assumed to affect the productivity of connected energy-producing "laborers". We show that collapse becomes increasingly likely as the complexity of the model society continuously increases in response to external stresses that emulate Tainter's abstract notion of problems that societies must solve. We also provide an analytical approximation of the system's dominant dynamics, which matches well with the numerical experiments, and use it to study the influence on network link density, social mobility and productivity. Our work advances the understanding of social-ecological collapse and illustrates its potentially direct link to an ever-increasing societal complexity in response to external shocks or stresses via a self-reinforcing feedback.
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Affiliation(s)
- Florian Schunck
- Research Group System Ecotox, Helmholtz Centre for Environmental Research GmbH-UFZ, Permoserstraße 15, 04318 Leipzig, Germany
- Research Group System Science, Institute of Mathematics, Osnabrück University, Barbarastraße 12, 49076 Osnabrück, Germany
| | - Marc Wiedermann
- FutureLab on Game Theory and Networks of Interacting Agents, FutureLab on Earth Resilience in the Anthropocene, Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
| | - Jobst Heitzig
- FutureLab on Game Theory and Networks of Interacting Agents, FutureLab on Earth Resilience in the Anthropocene, Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
| | - Jonathan F Donges
- FutureLab on Game Theory and Networks of Interacting Agents, FutureLab on Earth Resilience in the Anthropocene, Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Albanovägen 28, 106 91 Stockholm, Sweden
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2
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Søgaard Jørgensen P, Jansen REV, Avila Ortega DI, Wang-Erlandsson L, Donges JF, Österblom H, Olsson P, Nyström M, Lade SJ, Hahn T, Folke C, Peterson GD, Crépin AS. Evolution of the polycrisis: Anthropocene traps that challenge global sustainability. Philos Trans R Soc Lond B Biol Sci 2024; 379:20220261. [PMID: 37952617 PMCID: PMC10645130 DOI: 10.1098/rstb.2022.0261] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2023] [Accepted: 08/22/2023] [Indexed: 11/14/2023] Open
Abstract
The Anthropocene is characterized by accelerating change and global challenges of increasing complexity. Inspired by what some have called a polycrisis, we explore whether the human trajectory of increasing complexity and influence on the Earth system could become a form of trap for humanity. Based on an adaptation of the evolutionary traps concept to a global human context, we present results from a participatory mapping. We identify 14 traps and categorize them as either global, technology or structural traps. An assessment reveals that 12 traps (86%) could be in an advanced phase of trapping with high risk of hard-to-reverse lock-ins and growing risks of negative impacts on human well-being. Ten traps (71%) currently see growing trends in their indicators. Revealing the systemic nature of the polycrisis, we assess that Anthropocene traps often interact reinforcingly (45% of pairwise interactions), and rarely in a dampening fashion (3%). We end by discussing capacities that will be important for navigating these systemic challenges in pursuit of global sustainability. Doing so, we introduce evolvability as a unifying concept for such research between the sustainability and evolutionary sciences. This article is part of the theme issue 'Evolution and sustainability: gathering the strands for an Anthropocene synthesis'.
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Affiliation(s)
- Peter Søgaard Jørgensen
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Global Economic Dynamics and the Biosphere Programme, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
- Anthropocene Laboratory, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
| | - Raf E. V. Jansen
- Global Economic Dynamics and the Biosphere Programme, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
| | - Daniel I. Avila Ortega
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Global Economic Dynamics and the Biosphere Programme, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
| | - Lan Wang-Erlandsson
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Anthropocene Laboratory, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
- Potsdam Institute for Climate Impact Research, Member of the Leibnitz Association, 14473 Potsdam, Germany
| | - Jonathan F. Donges
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Potsdam Institute for Climate Impact Research, Member of the Leibnitz Association, 14473 Potsdam, Germany
| | - Henrik Österblom
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Anthropocene Laboratory, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
| | - Per Olsson
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Magnus Nyström
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Steven J. Lade
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Fenner School of Environment & Society, Australian National University, Canberra 2601, Australia
| | - Thomas Hahn
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Carl Folke
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Global Economic Dynamics and the Biosphere Programme, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
- Anthropocene Laboratory, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
- Beijer Institute of Ecological Economics, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
| | - Garry D. Peterson
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
| | - Anne-Sophie Crépin
- Stockholm Resilience Centre, Stockholm University, SE-106 91 Stockholm, Sweden
- Beijer Institute of Ecological Economics, Royal Swedish Academy of Sciences, SE-104 05 Stockholm, Sweden
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3
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Richardson K, Steffen W, Lucht W, Bendtsen J, Cornell SE, Donges JF, Drüke M, Fetzer I, Bala G, von Bloh W, Feulner G, Fiedler S, Gerten D, Gleeson T, Hofmann M, Huiskamp W, Kummu M, Mohan C, Nogués-Bravo D, Petri S, Porkka M, Rahmstorf S, Schaphoff S, Thonicke K, Tobian A, Virkki V, Wang-Erlandsson L, Weber L, Rockström J. Earth beyond six of nine planetary boundaries. Sci Adv 2023; 9:eadh2458. [PMID: 37703365 PMCID: PMC10499318 DOI: 10.1126/sciadv.adh2458] [Citation(s) in RCA: 54] [Impact Index Per Article: 54.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2023] [Accepted: 07/12/2023] [Indexed: 09/15/2023]
Abstract
This planetary boundaries framework update finds that six of the nine boundaries are transgressed, suggesting that Earth is now well outside of the safe operating space for humanity. Ocean acidification is close to being breached, while aerosol loading regionally exceeds the boundary. Stratospheric ozone levels have slightly recovered. The transgression level has increased for all boundaries earlier identified as overstepped. As primary production drives Earth system biosphere functions, human appropriation of net primary production is proposed as a control variable for functional biosphere integrity. This boundary is also transgressed. Earth system modeling of different levels of the transgression of the climate and land system change boundaries illustrates that these anthropogenic impacts on Earth system must be considered in a systemic context.
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Affiliation(s)
- Katherine Richardson
- Globe Institute, Faculty of Health, University of Copenhagen, Copenhagen, Denmark
| | - Will Steffen
- Australian National University, Canberra, Australia
| | - Wolfgang Lucht
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Department of Geography, Humboldt-Universität zu Berlin, Berlin, Germany
| | - Jørgen Bendtsen
- Globe Institute, Faculty of Health, University of Copenhagen, Copenhagen, Denmark
| | - Sarah E. Cornell
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
| | - Jonathan F. Donges
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
| | - Markus Drüke
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Ingo Fetzer
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
- Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden
| | - Govindasamy Bala
- Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore, Karnataka – 560012, India
| | - Werner von Bloh
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Georg Feulner
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Stephanie Fiedler
- GEOMAR Helmholtz Centre for Ocean Research Kiel and Faculty for Mathematics and Natural Sciences, Christian-Albrechts-University Kiel, Kiel, Germany
| | - Dieter Gerten
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Department of Geography, Humboldt-Universität zu Berlin, Berlin, Germany
| | - Tom Gleeson
- Department of Civil Engineering, University of Victoria, Victoria, British Columbia, Canada
- School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
| | - Matthias Hofmann
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Willem Huiskamp
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Matti Kummu
- Water and Development Research Group, Aalto University, Espoo, Finland
| | - Chinchu Mohan
- GEOMAR Helmholtz Centre for Ocean Research Kiel and Faculty for Mathematics and Natural Sciences, Christian-Albrechts-University Kiel, Kiel, Germany
- Global Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
- Waterplan (YC S21), San Francisco, CA, USA
| | - David Nogués-Bravo
- Globe Institute, Faculty of Health, University of Copenhagen, Copenhagen, Denmark
| | - Stefan Petri
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Miina Porkka
- Water and Development Research Group, Aalto University, Espoo, Finland
| | - Stefan Rahmstorf
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
| | - Sibyll Schaphoff
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Kirsten Thonicke
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
| | - Arne Tobian
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
| | - Vili Virkki
- Water and Development Research Group, Aalto University, Espoo, Finland
| | - Lan Wang-Erlandsson
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
- Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden
| | - Lisa Weber
- GEOMAR Helmholtz Centre for Ocean Research Kiel and Faculty for Mathematics and Natural Sciences, Christian-Albrechts-University Kiel, Kiel, Germany
| | - Johan Rockström
- Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
- Institute for Environmental Science and Geography, University of Potsdam, Potsdam, Germany
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4
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Bien S, Schultz P, Heitzig J, Donges JF. Resilience basins of complex systems: An application to prosumer impacts on power grids. Chaos 2023; 33:063148. [PMID: 37352506 DOI: 10.1063/5.0120891] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 06/01/2023] [Indexed: 06/25/2023]
Abstract
Comparable to the traditional notion of stability in system dynamics, resilience is typically measured in a way that assesses the quality of a system's response, for example, the speed of its recovery. We present a broadly applicable complementary measurement framework that quantifies resilience similarly to basin stability by estimating a resilience basin, which reflects the extent of adverse influences that the system can recover from in a sufficient manner. In contrast to basin stability, the adverse influences considered here are not necessarily displacements in state space, but arbitrarily complex impacts to the system, quantified by adequate parameters. As a proof of concept, we present two applications: (i) the well-studied single-node power system as an easy-to-follow example and (ii) a stochastic model of a low-voltage DC power grid undergoing an unregulated energy transition consisting in the random appearance of prosumers. These act as decentral suppliers of photovoltaic power and alter the flow patterns while the grid topology remains unchanged. The resilience measurement framework is applied to evaluate the effect and efficiency of two response options: (i) upgrading the capacity of existing power lines and (ii) installing batteries in the prosumer households. The framework demonstrates that line upgrades can provide potentially unlimited resilience against energy decentralization, while household batteries are inherently limited (achieving ≤70% of the resilience of line upgrades). Further, the framework aids in optimizing budget efficiency by pointing toward threshold budget values as well as budget-dependent ideal strategies for the allocation of line upgrades and for the battery charging algorithm.
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Affiliation(s)
- Samuel Bien
- Institute of Environmental Science and Geography, Potsdam University, 14469 Potsdam, Germany
- Institute of Physics, Potsdam University, 14469 Potsdam, Germany
- Complexity Science Department, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- FutureLab for Earth Resilience in the Anthropocene, Earth System Analysis Department, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Paul Schultz
- 50Hertz Transmission GmbH, 10557 Berlin, Germany
| | - Jobst Heitzig
- FutureLab for Game Theory and Networks of Interacting Agents, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Jonathan F Donges
- FutureLab for Earth Resilience in the Anthropocene, Earth System Analysis Department, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, 106 91 Stockholm, Sweden
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5
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Kohler J, Wunderling N, Donges JF, Vollmer J. Complex networks of interacting stochastic tipping elements: Cooperativity of phase separation in the large-system limit. Phys Rev E 2021; 104:044301. [PMID: 34781496 DOI: 10.1103/physreve.104.044301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2021] [Accepted: 08/26/2021] [Indexed: 11/07/2022]
Abstract
Tipping elements in the Earth system have received increased scientific attention over recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are subdominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks of different size and with different coupling parameters.
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Affiliation(s)
- Jan Kohler
- Institute for Theoretical Physics, University of Leipzig, 04103 Leipzig, Germany, EU.,Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU
| | - Nico Wunderling
- Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU.,Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany, EU.,Department of Physics, Humboldt University of Berlin, 12489 Berlin, Germany, EU
| | - Jonathan F Donges
- Earth System Analysis, Potsdam-Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany, EU.,Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden, EU
| | - Jürgen Vollmer
- Institute for Theoretical Physics, University of Leipzig, 04103 Leipzig, Germany, EU
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6
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Donges JF, Lochner JH, Kitzmann NH, Heitzig J, Lehmann S, Wiedermann M, Vollmer J. Dose-response functions and surrogate models for exploring social contagion in the Copenhagen Networks Study. Eur Phys J Spec Top 2021; 230:3311-3334. [PMID: 34611486 PMCID: PMC8484857 DOI: 10.1140/epjs/s11734-021-00279-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Accepted: 09/03/2021] [Indexed: 06/13/2023]
Abstract
Spreading dynamics and complex contagion processes on networks are important mechanisms underlying the emergence of critical transitions, tipping points and other non-linear phenomena in complex human and natural systems. Increasing amounts of temporal network data are now becoming available to study such spreading processes of behaviours, opinions, ideas, diseases and innovations to test hypotheses regarding their specific properties. To this end, we here present a methodology based on dose-response functions and hypothesis testing using surrogate data models that randomise most aspects of the empirical data while conserving certain structures relevant to contagion, group or homophily dynamics. We demonstrate this methodology for synthetic temporal network data of spreading processes generated by the adaptive voter model. Furthermore, we apply it to empirical temporal network data from the Copenhagen Networks Study. This data set provides a physically-close-contact network between several hundreds of university students participating in the study over the course of 3 months. We study the potential spreading dynamics of the health-related behaviour "regularly going to the fitness studio" on this network. Based on a hierarchy of surrogate data models, we find that our method neither provides significant evidence for an influence of a dose-response-type network spreading process in this data set, nor significant evidence for homophily. The empirical dynamics in exercise behaviour are likely better described by individual features such as the disposition towards the behaviour, and the persistence to maintain it, as well as external influences affecting the whole group, and the non-trivial network structure. The proposed methodology is generic and promising also for applications to other temporal network data sets and traits of interest.
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Affiliation(s)
- Jonathan F. Donges
- Earth System Analysis and Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
| | - Jakob H. Lochner
- Earth System Analysis and Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
- Institute for Theoretical Physics, University of Leipzig, Leipzig, Germany
| | - Niklas H. Kitzmann
- Earth System Analysis and Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
- Institute for Physics and Astronomy, University of Potsdam, Potsdam, Germany
| | - Jobst Heitzig
- Earth System Analysis and Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
| | - Sune Lehmann
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
- Center for Social Data Science, University of Copenhagen, Copenhagen, Denmark
| | - Marc Wiedermann
- Earth System Analysis and Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
- Robert Koch-Institut, Berlin, Germany
- Institute for Theoretical Biology, Humboldt University of Berlin, Berlin, Germany
| | - Jürgen Vollmer
- Institute for Theoretical Physics, University of Leipzig, Leipzig, Germany
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7
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Bak-Coleman JB, Alfano M, Barfuss W, Bergstrom CT, Centeno MA, Couzin ID, Donges JF, Galesic M, Gersick AS, Jacquet J, Kao AB, Moran RE, Romanczuk P, Rubenstein DI, Tombak KJ, Van Bavel JJ, Weber EU. Stewardship of global collective behavior. Proc Natl Acad Sci U S A 2021; 118:e2025764118. [PMID: 34155097 PMCID: PMC8271675 DOI: 10.1073/pnas.2025764118] [Citation(s) in RCA: 63] [Impact Index Per Article: 21.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
Collective behavior provides a framework for understanding how the actions and properties of groups emerge from the way individuals generate and share information. In humans, information flows were initially shaped by natural selection yet are increasingly structured by emerging communication technologies. Our larger, more complex social networks now transfer high-fidelity information over vast distances at low cost. The digital age and the rise of social media have accelerated changes to our social systems, with poorly understood functional consequences. This gap in our knowledge represents a principal challenge to scientific progress, democracy, and actions to address global crises. We argue that the study of collective behavior must rise to a "crisis discipline" just as medicine, conservation, and climate science have, with a focus on providing actionable insight to policymakers and regulators for the stewardship of social systems.
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Affiliation(s)
- Joseph B Bak-Coleman
- Center for an Informed Public, University of Washington, Seattle, WA 98195;
- eScience Institute, University of Washington, Seattle, WA 98195
| | - Mark Alfano
- Ethics & Philosophy of Technology, Delft University of Technology, 2628 CD Delft, The Netherlands
- Institute of Philosophy, Australian Catholic University, Banyo Queensland 4014, Australia
| | - Wolfram Barfuss
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany
- Tübingen AI Center, University of Tübingen, 72074 Tübingen, Germany
| | - Carl T Bergstrom
- Department of Biology, University of Washington, Seattle, WA 98195
| | - Miguel A Centeno
- Princeton School of Public and International Affairs, Princeton University, Princeton, NJ 08544
| | - Iain D Couzin
- Department of Collective Behaviour, Max Planck Institute of Animal Behavior, 78315 Radolfzell am Bodensee, Germany
- Centre for the Advanced Study of Collective Behaviour, University of Konstanz, 78464 Konstanz, Germany
- Department of Biology, University of Konstanz, 78464 Konstanz, Germany
| | - Jonathan F Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, 11419 Stockholm, Sweden
| | | | - Andrew S Gersick
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544
| | - Jennifer Jacquet
- Department of Environmental Studies, New York University, New York, NY 10012
| | | | - Rachel E Moran
- Center for an Informed Public, University of Washington, Seattle, WA 98195
| | - Pawel Romanczuk
- Institute for Theoretical Biology, Department of Biology, Humboldt Universität zu Berlin, 10115 Berlin, Germany
| | - Daniel I Rubenstein
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544
| | - Kaia J Tombak
- Department of Anthropology, Hunter College of the City University of New York, New York, NY 10065
| | - Jay J Van Bavel
- Department of Psychology, New York University, New York, NY 10003
- Center for Neural Science, New York University, New York, NY 10003
| | - Elke U Weber
- Department of Psychology, Princeton University, Princeton, NJ 08544
- Andlinger Center for Energy and Environment, School of Engineering and Applied Science, Princeton University, Princeton, NJ 08544
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8
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Wunderling N, Willeit M, Donges JF, Winkelmann R. Global warming due to loss of large ice masses and Arctic summer sea ice. Nat Commun 2020; 11:5177. [PMID: 33110092 PMCID: PMC7591863 DOI: 10.1038/s41467-020-18934-3] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2019] [Accepted: 09/17/2020] [Indexed: 11/09/2022] Open
Abstract
Several large-scale cryosphere elements such as the Arctic summer sea ice, the mountain glaciers, the Greenland and West Antarctic Ice Sheet have changed substantially during the last century due to anthropogenic global warming. However, the impacts of their possible future disintegration on global mean temperature (GMT) and climate feedbacks have not yet been comprehensively evaluated. Here, we quantify this response using an Earth system model of intermediate complexity. Overall, we find a median additional global warming of 0.43 °C (interquartile range: 0.39-0.46 °C) at a CO2 concentration of 400 ppm. Most of this response (55%) is caused by albedo changes, but lapse rate together with water vapour (30%) and cloud feedbacks (15%) also contribute significantly. While a decay of the ice sheets would occur on centennial to millennial time scales, the Arctic might become ice-free during summer within the 21st century. Our findings imply an additional increase of the GMT on intermediate to long time scales.
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Affiliation(s)
- Nico Wunderling
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, D-14473, Germany. .,Institute of Physics and Astronomy, University of Potsdam, Potsdam, D-14476, Germany. .,Department of Physics, Humboldt University of Berlin, Berlin, D-12489, Germany.
| | - Matteo Willeit
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, D-14473, Germany
| | - Jonathan F Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, D-14473, Germany.,Stockholm Resilience Centre, Stockholm University, Stockholm, SE, 10691, Sweden
| | - Ricarda Winkelmann
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, Potsdam, D-14473, Germany. .,Institute of Physics and Astronomy, University of Potsdam, Potsdam, D-14476, Germany.
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9
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Wiedermann M, Smith EK, Heitzig J, Donges JF. A network-based microfoundation of Granovetter's threshold model for social tipping. Sci Rep 2020; 10:11202. [PMID: 32641784 PMCID: PMC7343878 DOI: 10.1038/s41598-020-67102-6] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 06/02/2020] [Indexed: 11/11/2022] Open
Abstract
Social tipping, where minorities trigger larger populations to engage in collective action, has been suggested as one key aspect in addressing contemporary global challenges. Here, we refine Granovetter’s widely acknowledged theoretical threshold model of collective behavior as a numerical modelling tool for understanding social tipping processes and resolve issues that so far have hindered such applications. Based on real-world observations and social movement theory, we group the population into certain or potential actors, such that – in contrast to its original formulation – the model predicts non-trivial final shares of acting individuals. Then, we use a network cascade model to explain and analytically derive that previously hypothesized broad threshold distributions emerge if individuals become active via social interaction. Thus, through intuitive parameters and low dimensionality our refined model is adaptable to explain the likelihood of engaging in collective behavior where social-tipping-like processes emerge as saddle-node bifurcations and hysteresis.
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Affiliation(s)
- Marc Wiedermann
- FutureLab on Game Theory & Networks of Interacting Agents, Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, P.O. Box 60 12 03, 14412, Potsdam, Germany.
| | - E Keith Smith
- GESIS - Leibniz Institute for the Social Sciences, Member of the Leibniz Association, Unter Sachsenhausen 6-8, 50667, Cologne, Germany.,Institute of Science, Technology and Policy, ETH Zurich, Zurich, Switzerland
| | - Jobst Heitzig
- FutureLab on Game Theory & Networks of Interacting Agents, Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, P.O. Box 60 12 03, 14412, Potsdam, Germany
| | - Jonathan F Donges
- FutureLab Earth Resilience in the Anthropocene, Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, P.O. Box 60 12 03, 14412, Potsdam, Germany.,Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19, Stockholm, Sweden
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10
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Klose AK, Karle V, Winkelmann R, Donges JF. Emergence of cascading dynamics in interacting tipping elements of ecology and climate. R Soc Open Sci 2020; 7:200599. [PMID: 32742700 PMCID: PMC7353982 DOI: 10.1098/rsos.200599] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2020] [Accepted: 05/18/2020] [Indexed: 05/08/2023]
Abstract
In ecology, climate and other fields, (sub)systems have been identified that can transition into a qualitatively different state when a critical threshold or tipping point in a driving process is crossed. An understanding of those tipping elements is of great interest given the increasing influence of humans on the biophysical Earth system. Complex interactions exist between tipping elements, e.g. physical mechanisms connect subsystems of the climate system. Based on earlier work on such coupled nonlinear systems, we systematically assessed the qualitative long-term behaviour of interacting tipping elements. We developed an understanding of the consequences of interactions on the tipping behaviour allowing for tipping cascades to emerge under certain conditions. The (narrative) application of these qualitative results to real-world examples of interacting tipping elements indicates that tipping cascades with profound consequences may occur: the interacting Greenland ice sheet and thermohaline ocean circulation might tip before the tipping points of the isolated subsystems are crossed. The eutrophication of the first lake in a lake chain might propagate through the following lakes without a crossing of their individual critical nutrient input levels. The possibility of emerging cascading tipping dynamics calls for the development of a unified theory of interacting tipping elements and the quantitative analysis of interacting real-world tipping elements.
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Affiliation(s)
- Ann Kristin Klose
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, 14473 Potsdam, Germany
- Carl von Ossietzky University Oldenburg, Oldenburg, Germany
| | - Volker Karle
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, 14473 Potsdam, Germany
- Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
| | - Ricarda Winkelmann
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, 14473 Potsdam, Germany
- Department of Physics and Astronomy, University of Potsdam, 14469 Potsdam, Germany
| | - Jonathan F. Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, 14473 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
- Author for correspondence: Jonathan F. Donges e-mail:
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11
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Krönke J, Wunderling N, Winkelmann R, Staal A, Stumpf B, Tuinenburg OA, Donges JF. Dynamics of tipping cascades on complex networks. Phys Rev E 2020; 101:042311. [PMID: 32422827 DOI: 10.1103/physreve.101.042311] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Accepted: 03/18/2020] [Indexed: 01/02/2023]
Abstract
Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy, and engineering. Tipping points are critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change of the system. Many systems with tipping points can be modeled as networks of coupled multistable subsystems, e.g., coupled patches of vegetation, connected lakes, interacting climate tipping elements, and multiscale infrastructure systems. In such networks, tipping events in one subsystem are able to induce tipping cascades via domino effects. Here, we investigate the effects of network topology on the occurrence of such cascades. Numerical cascade simulations with a conceptual dynamical model for tipping points are conducted on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert networks. Additionally, we generate more realistic networks using data from moisture-recycling simulations of the Amazon rainforest and compare the results to those obtained for the model networks. We furthermore use a directed configuration model and a stochastic block model which preserve certain topological properties of the Amazon network to understand which of these properties are responsible for its increased vulnerability. We find that clustering and spatial organization increase the vulnerability of networks and can lead to tipping of the whole network. These results could be useful to evaluate which systems are vulnerable or robust due to their network topology and might help us to design or manage systems accordingly.
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Affiliation(s)
- Jonathan Krönke
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany.,Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
| | - Nico Wunderling
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany.,Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany.,Department of Physics, Humboldt University of Berlin, 12489 Berlin, Germany
| | - Ricarda Winkelmann
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany.,Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
| | - Arie Staal
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
| | - Benedikt Stumpf
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany.,Department of Physics, Free University Berlin, 14195 Berlin, Germany
| | - Obbe A Tuinenburg
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden.,Copernicus Institute, Faculty of Geosciences, Utrecht University, 3584 CB Utrecht, The Netherlands
| | - Jonathan F Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany.,Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
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12
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Otto IM, Donges JF, Lucht W, Schellnhuber HJ. Reply to Smith et al.: Social tipping dynamics in a world constrained by conflicting interests. Proc Natl Acad Sci U S A 2020; 117:10631-10632. [PMID: 32327604 PMCID: PMC7245088 DOI: 10.1073/pnas.2002648117] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Affiliation(s)
- Ilona M Otto
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany;
| | - Jonathan F Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, 11419 Stockholm, Sweden
| | - Wolfgang Lucht
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany
- Department of Geography, Humboldt University, 10099 Berlin, Germany
- Integrative Research Institute on Transformations of Human-Environment Systems, Humboldt University, 10099 Berlin, Germany
| | - Hans Joachim Schellnhuber
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, 14473 Potsdam, Germany
- Department of Earth System Science, School of Science, Tsinghua University, Beijing 100084, People's Republic of China
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13
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Wunderling N, Stumpf B, Krönke J, Staal A, Tuinenburg OA, Winkelmann R, Donges JF. How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales. Chaos 2020; 30:043129. [PMID: 32357654 DOI: 10.1063/1.5142827] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2019] [Accepted: 04/02/2020] [Indexed: 05/24/2023]
Abstract
In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős-Rényi network. This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole.
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Affiliation(s)
- Nico Wunderling
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany
| | - Benedikt Stumpf
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany
| | - Jonathan Krönke
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany
| | - Arie Staal
- Stockholm Resilience Centre, Stockholm University, Stockholm SE-10691, Sweden
| | - Obbe A Tuinenburg
- Stockholm Resilience Centre, Stockholm University, Stockholm SE-10691, Sweden
| | - Ricarda Winkelmann
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany
| | - Jonathan F Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, 14473 Potsdam, Germany
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14
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Naidu PD, Ganeshram R, Bollasina MA, Panmei C, Nürnberg D, Donges JF. Coherent response of the Indian Monsoon Rainfall to Atlantic Multi-decadal Variability over the last 2000 years. Sci Rep 2020; 10:1302. [PMID: 31992786 PMCID: PMC6987308 DOI: 10.1038/s41598-020-58265-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2019] [Accepted: 12/21/2019] [Indexed: 11/14/2022] Open
Abstract
Indian Summer Monsoon (ISM) rainfall has a direct effect on the livelihoods of two billion people in the Indian-subcontinent. Yet, our understanding of the drivers of multi-decadal variability of the ISM is far from being complete. In this context, large-scale forcing of ISM rainfall variability with multi-decadal resolution over the last two millennia is investigated using new records of sea surface salinity (δ18Ow) and sea surface temperatures (SSTs) from the Bay of Bengal (BoB). Higher δ18Ow values during the Dark Age Cold Period (1550 to 1250 years BP) and the Little Ice Age (700 to 200 years BP) are suggestive of reduced ISM rainfall, whereas lower δ18Ow values during the Medieval Warm Period (1200 to 800 years BP) and the major portion of the Roman Warm Period (1950 to 1550 years BP) indicate a wetter ISM. This variability in ISM rainfall appears to be modulated by the Atlantic Multi-decadal Oscillation (AMO) via changes in large-scale thermal contrast between the Asian land mass and the Indian Ocean, a relationship that is also identifiable in the observational data of the last century. Therefore, we suggest that inter-hemispheric scale interactions between such extra tropical forcing mechanisms and global warming are likely to be influential in determining future trends in ISM rainfall.
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Affiliation(s)
| | - Raja Ganeshram
- School of Geosciences, University of Edinburgh, Edinburgh, UK
| | | | - Champoungam Panmei
- CSIR-National Institute of Oceanography, Dona Paula, 403004, Goa, India
- Academy of Scientific and Innovative Research (AcSIR), CSIR-NIO, Goa, India
| | | | - Jonathan F Donges
- Postdam Institute for Climate Impact Research, P.O. Box 601203, D-14412, Postdam, Germany
- Planetary Boundary Research Lab, Stockholm Resilience Center, Stockholm University, Stockholm, Sweden
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15
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Strnad FM, Barfuss W, Donges JF, Heitzig J. Deep reinforcement learning in World-Earth system models to discover sustainable management strategies. Chaos 2019; 29:123122. [PMID: 31893656 DOI: 10.1063/1.5124673] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2019] [Accepted: 11/20/2019] [Indexed: 06/10/2023]
Abstract
Increasingly complex nonlinear World-Earth system models are used for describing the dynamics of the biophysical Earth system and the socioeconomic and sociocultural World of human societies and their interactions. Identifying pathways toward a sustainable future in these models for informing policymakers and the wider public, e.g., pathways leading to robust mitigation of dangerous anthropogenic climate change, is a challenging and widely investigated task in the field of climate research and broader Earth system science. This problem is particularly difficult when constraints on avoiding transgressions of planetary boundaries and social foundations need to be taken into account. In this work, we propose to combine recently developed machine learning techniques, namely, deep reinforcement learning (DRL), with classical analysis of trajectories in the World-Earth system. Based on the concept of the agent-environment interface, we develop an agent that is generally able to act and learn in variable manageable environment models of the Earth system. We demonstrate the potential of our framework by applying DRL algorithms to two stylized World-Earth system models. Conceptually, we explore thereby the feasibility of finding novel global governance policies leading into a safe and just operating space constrained by certain planetary and socioeconomic boundaries. The artificially intelligent agent learns that the timing of a specific mix of taxing carbon emissions and subsidies on renewables is of crucial relevance for finding World-Earth system trajectories that are sustainable in the long term.
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Affiliation(s)
- Felix M Strnad
- FutureLab on Game Theory and Networks of Interacting Agents, Research Department 4: Complexity Science, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Wolfram Barfuss
- FutureLab on Earth Resilience in the Anthropocene, Research Department 1: Earth System Analysis, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Jonathan F Donges
- FutureLab on Earth Resilience in the Anthropocene, Research Department 1: Earth System Analysis, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Jobst Heitzig
- FutureLab on Game Theory and Networks of Interacting Agents, Research Department 4: Complexity Science, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
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16
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Barfuss W, Donges JF, Kurths J. Deterministic limit of temporal difference reinforcement learning for stochastic games. Phys Rev E 2019; 99:043305. [PMID: 31108579 DOI: 10.1103/physreve.99.043305] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2018] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Wolfram Barfuss
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany.,Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany.,Stockholm Resilience Centre, Stockholm University, 104 05 Stockholm, Sweden
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany.,Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany.,Saratov State University, 410012 Saratov, Russia
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17
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Steffen W, Rockström J, Richardson K, Lenton TM, Folke C, Liverman D, Summerhayes CP, Barnosky AD, Cornell SE, Crucifix M, Donges JF, Fetzer I, Lade SJ, Scheffer M, Winkelmann R, Schellnhuber HJ. Trajectories of the Earth System in the Anthropocene. Proc Natl Acad Sci U S A 2018; 115:8252-8259. [PMID: 30082409 PMCID: PMC6099852 DOI: 10.1073/pnas.1810141115] [Citation(s) in RCA: 435] [Impact Index Per Article: 72.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
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.
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Affiliation(s)
- Will Steffen
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden;
- Fenner School of Environment and Society, The Australian National University, Canberra, ACT 2601, Australia
| | - Johan Rockström
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
| | - Katherine Richardson
- Center for Macroecology, Evolution, and Climate, University of Copenhagen, Natural History Museum of Denmark, 2100 Copenhagen, Denmark
| | - Timothy M Lenton
- Earth System Science Group, College of Life and Environmental Sciences, University of Exeter, EX4 4QE Exeter, United Kingdom
| | - Carl Folke
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
- The Beijer Institute of Ecological Economics, The Royal Swedish Academy of Science, SE-10405 Stockholm, Sweden
| | - Diana Liverman
- School of Geography and Development, The University of Arizona, Tucson, AZ 85721
| | - Colin P Summerhayes
- Scott Polar Research Institute, Cambridge University, CB2 1ER Cambridge, United Kingdom
| | - Anthony D Barnosky
- Jasper Ridge Biological Preserve, Stanford University, Stanford, CA 94305
| | - Sarah E Cornell
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
| | - Michel Crucifix
- Earth and Life Institute, Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
- Belgian National Fund of Scientific Research, 1000 Brussels, Belgium
| | - Jonathan F Donges
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
- Research Domain Earth System Analysis, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
| | - Ingo Fetzer
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
| | - Steven J Lade
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden
- Fenner School of Environment and Society, The Australian National University, Canberra, ACT 2601, Australia
| | - Marten Scheffer
- Department of Environmental Sciences, Wageningen University & Research, 6700AA Wageningen, The Netherlands
| | - Ricarda Winkelmann
- Research Domain Earth System Analysis, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- Department of Physics and Astronomy, University of Potsdam, 14469 Potsdam, Germany
| | - Hans Joachim Schellnhuber
- Stockholm Resilience Centre, Stockholm University, 10691 Stockholm, Sweden;
- Research Domain Earth System Analysis, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- Department of Physics and Astronomy, University of Potsdam, 14469 Potsdam, Germany
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18
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Donner RV, Stolbova V, Balasis G, Donges JF, Georgiou M, Potirakis SM, Kurths J. Temporal organization of magnetospheric fluctuations unveiled by recurrence patterns in the Dst index. Chaos 2018; 28:085716. [PMID: 30180615 DOI: 10.1063/1.5024792] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2018] [Accepted: 06/22/2018] [Indexed: 06/08/2023]
Abstract
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.
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Affiliation(s)
- Reik V Donner
- Research Domain IV-Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Veronika Stolbova
- Research Domain IV-Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Georgios Balasis
- Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, I. Metaxa & Vas. Pavlou Street, 15236 Penteli, Greece
| | - Jonathan F Donges
- Research Domain I-Earth System Analysis, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Marina Georgiou
- Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, I. Metaxa & Vas. Pavlou Street, 15236 Penteli, Greece
| | - Stelios M Potirakis
- Department of Electrical and Electronics Engineering, University of West Attica, Campus 2, 250 Thivon and P. Ralli, Aigaleo, 12244 Athens, Greece
| | - Jürgen Kurths
- Research Domain IV-Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
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19
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Barfuss W, Donges JF, Lade SJ, Kurths J. When optimization for governing human-environment tipping elements is neither sustainable nor safe. Nat Commun 2018; 9:2354. [PMID: 29907743 PMCID: PMC6003916 DOI: 10.1038/s41467-018-04738-z] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2018] [Accepted: 05/16/2018] [Indexed: 11/09/2022] Open
Abstract
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.
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Affiliation(s)
- Wolfram Barfuss
- Potsdam Institute for Climate Impact Research, 14473, Potsdam, Germany.
- Department of Physics, Humboldt University, 12489, Berlin, Germany.
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, 14473, Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, 11419, Stockholm, Sweden
| | - Steven J Lade
- Stockholm Resilience Centre, Stockholm University, 11419, Stockholm, Sweden
- Fenner School of Environment and Society, The Australian National University, Canberra, ACT, 2601, Australia
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, 14473, Potsdam, Germany
- Department of Physics, Humboldt University, 12489, Berlin, Germany
- Saratov State University, Saratov, 410012, Russia
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20
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Abstract
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.
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Affiliation(s)
- Pascal P Klamser
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.,Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany
| | - Marc Wiedermann
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.,Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.,Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden
| | - Reik V Donner
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany
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21
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Abstract
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.
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Affiliation(s)
- Marc Wiedermann
- Potsdam Institute for Climate Impact Research, Telegraphenberg A31, 14473 Potsdam, Germany, EU
- Department of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany, EU
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, Telegraphenberg A31, 14473 Potsdam, Germany, EU
- Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden, EU
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Telegraphenberg A31, 14473 Potsdam, Germany, EU
- Department of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany, EU
| | - Reik V Donner
- Potsdam Institute for Climate Impact Research, Telegraphenberg A31, 14473 Potsdam, Germany, EU
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22
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Fujiwara N, Kirchen K, Donges JF, Donner RV. A perturbation-theoretic approach to Lagrangian flow networks. Chaos 2017; 27:035813. [PMID: 28364772 DOI: 10.1063/1.4978549] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2016] [Accepted: 02/28/2017] [Indexed: 06/07/2023]
Abstract
Complex network approaches have been successfully applied for studying transport processes in complex systems ranging from road, railway, or airline infrastructures over industrial manufacturing to fluid dynamics. Here, we utilize a generic framework for describing the dynamics of geophysical flows such as ocean currents or atmospheric wind fields in terms of Lagrangian flow networks. In this approach, information on the passive advection of particles is transformed into a Markov chain based on transition probabilities of particles between the volume elements of a given partition of space for a fixed time step. We employ perturbation-theoretic methods to investigate the effects of modifications of transport processes in the underlying flow for three different problem classes: efficient absorption (corresponding to particle trapping or leaking), constant input of particles (with additional source terms modeling, e.g., localized contamination), and shifts of the steady state under probability mass conservation (as arising if the background flow is perturbed itself). Our results demonstrate that in all three cases, changes to the steady state solution can be analytically expressed in terms of the eigensystem of the unperturbed flow and the perturbation itself. These results are potentially relevant for developing more efficient strategies for coping with contaminations of fluid or gaseous media such as ocean and atmosphere by oil spills, radioactive substances, non-reactive chemicals, or volcanic aerosols.
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Affiliation(s)
- Naoya Fujiwara
- Center for Spatial Information Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kahshiwa-shi, Chiba 277-8568, Japan
| | - Kathrin Kirchen
- Research Domain IV-Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Jonathan F Donges
- Research Domain I-Earth System Analysis, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Reik V Donner
- Research Domain IV-Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
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23
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Molkenthin N, Kutza H, Tupikina L, Marwan N, Donges JF, Feudel U, Kurths J, Donner RV. Edge anisotropy and the geometric perspective on flow networks. Chaos 2017; 27:035802. [PMID: 28364754 DOI: 10.1063/1.4971785] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Spatial networks have recently attracted great interest in various fields of research. While the traditional network-theoretic viewpoint is commonly restricted to their topological characteristics (often disregarding 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 spatio-temporal data, which is complementary to topological characteristics of the same flow networks. Taken both structural and geometric viewpoints together can thus assist the identification of underlying flow structures from observations of scalar variables.
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Affiliation(s)
- Nora Molkenthin
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Hannes Kutza
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Liubov Tupikina
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Norbert Marwan
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Jonathan F Donges
- Research Domain I - Earth System Analysis, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Ulrike Feudel
- Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University, Carl-von-Ossietzky-Straße 9, 26129 Oldenburg, Germany
| | - Jürgen Kurths
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
| | - Reik V Donner
- Research Domain IV - Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, Telegrafenberg A31, 14473 Potsdam, Germany
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24
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Schleussner CF, Donges JF, Engemann DA, Levermann A. Clustered marginalization of minorities during social transitions induced by co-evolution of behaviour and network structure. Sci Rep 2016; 6:30790. [PMID: 27510641 PMCID: PMC4980617 DOI: 10.1038/srep30790] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2016] [Accepted: 07/11/2016] [Indexed: 11/09/2022] Open
Abstract
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.
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Affiliation(s)
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, Potsdam, Germany.,Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden
| | - Denis A Engemann
- Cognitive Neuroimaging Unit, CEA DRF/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, 91191 Gif/Yvette, France.,Neuropsychology &Neuroimaging Team, INSERM UMRS 975, ICM, Paris, France
| | - Anders Levermann
- Potsdam Institute for Climate Impact Research, Potsdam, Germany.,Lamont-Doherty Earth Observatory, Columbia University, New York, USA.,Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
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25
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Wiedermann M, Donges JF, Kurths J, Donner RV. Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes. Phys Rev E 2016; 93:042308. [PMID: 27176313 DOI: 10.1103/physreve.93.042308] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2015] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Marc Wiedermann
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany, EU.,Department of Physics, Humboldt University, Newtonstraße 15, 12489 Berlin, Germany, EU
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany, EU.,Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden, EU
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany, EU.,Department of Physics, Humboldt University, Newtonstraße 15, 12489 Berlin, Germany, EU.,Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom, EU.,Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia
| | - Reik V Donner
- Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany, EU
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26
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van Kan A, Jegminat J, Donges JF, Kurths J. Constrained basin stability for studying transient phenomena in dynamical systems. Phys Rev E 2016; 93:042205. [PMID: 27176291 DOI: 10.1103/physreve.93.042205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2015] [Indexed: 06/05/2023]
Abstract
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.
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Affiliation(s)
- Adrian van Kan
- Department of Physics and Astronomy, University of Heidelberg, Im Neuenheimer Feld 226, D-69120 Heidelberg, Germany
- Department of Physics, Imperial College London, Prince Consort Rd, London SW7 2BB, United Kingdom
| | - Jannes Jegminat
- Department of Physics and Astronomy, University of Heidelberg, Im Neuenheimer Feld 226, D-69120 Heidelberg, Germany
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
- Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
- Department of Physics, Humboldt University Berlin, Newtonstr. 15, D-12489 Berlin, Germany
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia
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27
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Donges JF, Heitzig J, Beronov B, Wiedermann M, Runge J, Feng QY, Tupikina L, Stolbova V, Donner RV, Marwan N, Dijkstra HA, Kurths J. Unified functional network and nonlinear time series analysis for complex systems science: The pyunicorn package. Chaos 2015; 25:113101. [PMID: 26627561 DOI: 10.1063/1.4934554] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We introduce the pyunicorn (Pythonic unified complex network and recurrence analysis toolbox) open source software package for applying and combining modern methods of data analysis and modeling from complex network theory and nonlinear time series analysis. pyunicorn is a fully object-oriented and easily parallelizable package written in the language Python. It allows for the construction of functional networks such as climate networks in climatology or functional brain networks in neuroscience representing the structure of statistical interrelationships in large data sets of time series and, subsequently, investigating this structure using advanced methods of complex network theory such as measures and models for spatial networks, networks of interacting networks, node-weighted statistics, or network surrogates. Additionally, pyunicorn provides insights into the nonlinear dynamics of complex systems as recorded in uni- and multivariate time series from a non-traditional perspective by means of recurrence quantification analysis, recurrence networks, visibility graphs, and construction of surrogate time series. The range of possible applications of the library is outlined, drawing on several examples mainly from the field of climatology.
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Affiliation(s)
- Jonathan F Donges
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Jobst Heitzig
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Boyan Beronov
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Marc Wiedermann
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Jakob Runge
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Qing Yi Feng
- Institute for Marine and Atmospheric Research Utrecht (IMAU), Department of Physics and Astronomy, Utrecht University, Utrecht, The Netherlands
| | - Liubov Tupikina
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Veronika Stolbova
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Reik V Donner
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Norbert Marwan
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
| | - Henk A Dijkstra
- Institute for Marine and Atmospheric Research Utrecht (IMAU), Department of Physics and Astronomy, Utrecht University, Utrecht, The Netherlands
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
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28
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Wiedermann M, Donges JF, Heitzig J, Lucht W, Kurths J. Macroscopic description of complex adaptive networks coevolving with dynamic node states. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 91:052801. [PMID: 26066206 DOI: 10.1103/physreve.91.052801] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2014] [Indexed: 06/04/2023]
Abstract
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.
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Affiliation(s)
- Marc Wiedermann
- Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany, EU
- Department of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany, EU
| | - Jonathan F Donges
- Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany, EU
- Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden, EU
| | - Jobst Heitzig
- Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany, EU
| | - Wolfgang Lucht
- Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany, EU
- Department of Geography, Humboldt University, Rudower Chaussee 16, 12489 Berlin, Germany, EU
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany, EU
- Department of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany, EU
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom, EU
- Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia
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29
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Lange S, Donges JF, Volkholz J, Kurths J. Local difference measures between complex networks for dynamical system model evaluation. PLoS One 2015; 10:e0118088. [PMID: 25856374 PMCID: PMC4391794 DOI: 10.1371/journal.pone.0118088] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2014] [Accepted: 01/04/2015] [Indexed: 11/23/2022] Open
Abstract
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. [8] we comparatively analyze statistical and dynamical regional climate simulations of the South American monsoon system [corrected]. 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.
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Affiliation(s)
- Stefan Lange
- Department of Physics, Humboldt University, Berlin, Germany
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - Jonathan F. Donges
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
- Stockholm Resilience Center, Stockholm University, Stockholm, Sweden
| | - Jan Volkholz
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - Jürgen Kurths
- Department of Physics, Humboldt University, Berlin, Germany
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen, United Kingdom
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30
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Radebach A, Donner RV, Runge J, Donges JF, Kurths J. Disentangling different types of El Niño episodes by evolving climate network analysis. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 88:052807. [PMID: 24329318 DOI: 10.1103/physreve.88.052807] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2011] [Revised: 07/19/2013] [Indexed: 06/03/2023]
Abstract
Complex network theory provides a powerful toolbox for studying the structure of statistical interrelationships between multiple time series in various scientific disciplines. In this work, we apply the recently proposed climate network approach for characterizing the evolving correlation structure of the Earth's climate system based on reanalysis data for surface air temperatures. We provide a detailed study of the temporal variability of several global climate network characteristics. Based on a simple conceptual view of red climate networks (i.e., networks with a comparably low number of edges), we give a thorough interpretation of our evolving climate network characteristics, which allows a functional discrimination between recently recognized different types of El Niño episodes. Our analysis provides deep insights into the Earth's climate system, particularly its global response to strong volcanic eruptions and large-scale impacts of different phases of the El Niño Southern Oscillation.
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31
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Donges JF, Heitzig J, Donner RV, Kurths J. Analytical framework for recurrence network analysis of time series. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:046105. [PMID: 22680536 DOI: 10.1103/physreve.85.046105] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2011] [Indexed: 05/27/2023]
Abstract
Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. While there are already many successful applications ranging from medicine to paleoclimatology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an ɛ-recurrence network as a discrete subnetwork of a "continuous" graph with uncountably many vertices and edges corresponding to the system's attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ɛ. In particular, we introduce local measures such as the ɛ-clustering coefficient, mesoscopic measures such as ɛ-motif density, path-based measures such as ɛ-betweennesses, and global measures such as ɛ-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ɛ via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasiperiodic motions, and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments. More generally, the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints, which appear frequently in disciplines ranging from biology to climate science.
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32
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Zou Y, Donner RV, Donges JF, Marwan N, Kurths J. Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods. Chaos 2010; 20:043130. [PMID: 21198100 DOI: 10.1063/1.3523304] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Rössler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.
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Affiliation(s)
- Yong Zou
- Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany
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33
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Donner RV, Zou Y, Donges JF, Marwan N, Kurths J. Ambiguities in recurrence-based complex network representations of time series. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:015101. [PMID: 20365421 DOI: 10.1103/physreve.81.015101] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2009] [Indexed: 05/29/2023]
Abstract
Recently, different approaches have been proposed for studying basic properties of time series from a complex network perspective. In this work, the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail. We discuss the main requirements that permit a feasible system-theoretic interpretation of network topology in terms of dynamically invariant phase-space properties. Possible artifacts induced by disregarding these requirements are pointed out and systematically studied. Finally, a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed.
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Affiliation(s)
- Reik V Donner
- Max Planck Institute for Physics of Complex Systems, Dresden, Germany.
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