The hysteresis of the Antarctic Ice Sheet.

More than half of Earth's freshwater resources are held by the Antarctic Ice Sheet, which thus represents by far the largest potential source for global sea-level rise under future warming conditions1. Its long-term stability determines the fate of our coastal cities and cultural heritage. Feedbacks between ice, atmosphere, ocean, and the solid Earth give rise to potential nonlinearities in its response to temperature changes. So far, we are lacking a comprehensive stability analysis of the Antarctic Ice Sheet for different amounts of global warming. Here we show that the Antarctic Ice Sheet exhibits a multitude of temperature thresholds beyond which ice loss is irreversible. Consistent with palaeodata2 we find, using the Parallel Ice Sheet Model3-5, that at global warming levels around 2 degrees Celsius above pre-industrial levels, West Antarctica is committed to long-term partial collapse owing to the marine ice-sheet instability. Between 6 and 9 degrees of warming above pre-industrial levels, the loss of more than 70 per cent of the present-day ice volume is triggered, mainly caused by the surface elevation feedback. At more than 10 degrees of warming above pre-industrial levels, Antarctica is committed to become virtually ice-free. The ice sheet's temperature sensitivity is 1.3 metres of sea-level equivalent per degree of warming up to 2 degrees above pre-industrial levels, almost doubling to 2.4 metres per degree of warming between 2 and 6 degrees and increasing to about 10 metres per degree of warming between 6 and 9 degrees. Each of these thresholds gives rise to hysteresis behaviour: that is, the currently observed ice-sheet configuration is not regained even if temperatures are reversed to present-day levels. In particular, the West Antarctic Ice Sheet does not regrow to its modern extent until temperatures are at least one degree Celsius lower than pre-industrial levels. Our results show that if the Paris Agreement is not met, Antarctica's long-term sea-level contribution will dramatically increase and exceed that of all other sources.

Recurrent droughts increase risk of cascading tipping events by outpacing adaptive capacities in the Amazon rainforest.

Tipping elements are nonlinear subsystems of the Earth system that have the potential to abruptly shift to another state if environmental change occurs close to a critical threshold with large consequences for human societies and ecosystems. Among these tipping elements may be the Amazon rainforest, which has been undergoing intensive anthropogenic activities and increasingly frequent droughts. Here, we assess how extreme deviations from climatological rainfall regimes may cause local forest collapse that cascades through the coupled forest-climate system. We develop a conceptual dynamic network model to isolate and uncover the role of atmospheric moisture recycling in such tipping cascades. We account for heterogeneity in critical thresholds of the forest caused by adaptation to local climatic conditions. Our results reveal that, despite this adaptation, a future climate characterized by permanent drought conditions could trigger a transition to an open canopy state particularly in the southern Amazon. The loss of atmospheric moisture recycling contributes to one-third of the tipping events. Thus, by exceeding local thresholds in forest adaptive capacity, local climate change impacts may propagate to other regions of the Amazon basin, causing a risk of forest shifts even in regions where critical thresholds have not been crossed locally.

The microdynamics of spatial polarization: A model and an application to survey data from Ukraine.

Although spatial polarization of attitudes is extremely common around the world, we understand little about the mechanisms through which polarization on divisive issues rises and falls over time. We develop a theory that explains how political shocks can have different effects in different regions of a country depending upon local dynamics generated by the preexisting spatial distribution of attitudes and discussion networks. Where opinions were previously divided, attitudinal diversity is likely to persist after the shock. Meanwhile, where a clear precrisis majority exists on key issues, opinions should change in the direction of the predominant view. These dynamics result in greater local homogeneity in attitudes but at the same time exacerbate geographic polarization across regions and sometimes even within regions. We illustrate our theory by developing a modified version of the adaptive voter model, an adaptive network model of opinion dynamics, to study changes in attitudes toward the European Union (EU) in Ukraine in the context of the Euromaidan Revolution of 2013 to 2014. Using individual-level panel data from surveys fielded before and after the Euromaidan Revolution, we show that EU support increased in areas with high prior public support for EU integration but declined further where initial public attitudes were opposed to the EU, thereby increasing the spatial polarization of EU attitudes in Ukraine. Our tests suggest that the predictive power of both network and regression models increases significantly when we incorporate information about the geographic location of network participants, which highlights the importance of spatially rooted social networks.

Stewardship of global collective behavior.

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.

A Dynamic Network Model of Societal Complexity and Resilience Inspired by Tainter's Theory of Collapse.

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.

Caring for the future can turn tragedy into comedy for long-term collective action under risk of collapse.

We will need collective action to avoid catastrophic climate change, and this will require valuing the long term as well as the short term. Shortsightedness and uncertainty have hindered progress in resolving this collective action problem and have been recognized as important barriers to cooperation among humans. Here, we propose a coupled social-ecological dilemma to investigate the interdependence of three well-identified components of this cooperation problem: 1) timescales of collapse and recovery in relation to time preferences regarding future outcomes, 2) the magnitude of the impact of collapse, and 3) the number of actors in the collective. We find that, under a sufficiently severe and time-distant collapse, how much the actors care for the future can transform the game from a tragedy of the commons into one of coordination, and even into a comedy of the commons in which cooperation dominates. Conversely, we also find conditions under which even strong concern for the future still does not transform the problem from tragedy to comedy. For a large number of participating actors, we find that the critical collapse impact, at which these game regime changes happen, converges to a fixed value of collapse impact per actor that is independent of the enhancement factor of the public good, which is usually regarded as the driver of the dilemma. Our results not only call for experimental testing but also help explain why polarization in beliefs about human-caused climate change can threaten global cooperation agreements.

Social tipping dynamics for stabilizing Earth's climate by 2050.

Safely achieving the goals of the Paris Climate Agreement requires a worldwide transformation to carbon-neutral societies within the next 30 y. Accelerated technological progress and policy implementations are required to deliver emissions reductions at rates sufficiently fast to avoid crossing dangerous tipping points in the Earth's climate system. Here, we discuss and evaluate the potential of social tipping interventions (STIs) that can activate contagious processes of rapidly spreading technologies, behaviors, social norms, and structural reorganization within their functional domains that we refer to as social tipping elements (STEs). STEs are subdomains of the planetary socioeconomic system where the required disruptive change may take place and lead to a sufficiently fast reduction in anthropogenic greenhouse gas emissions. The results are based on online expert elicitation, a subsequent expert workshop, and a literature review. The STIs that could trigger the tipping of STE subsystems include 1) removing fossil-fuel subsidies and incentivizing decentralized energy generation (STE1, energy production and storage systems), 2) building carbon-neutral cities (STE2, human settlements), 3) divesting from assets linked to fossil fuels (STE3, financial markets), 4) revealing the moral implications of fossil fuels (STE4, norms and value systems), 5) strengthening climate education and engagement (STE5, education system), and 6) disclosing information on greenhouse gas emissions (STE6, information feedbacks). Our research reveals important areas of focus for larger-scale empirical and modeling efforts to better understand the potentials of harnessing social tipping dynamics for climate change mitigation.

Resilience basins of complex systems: An application to prosumer impacts on power grids.

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.

Trajectories of the Earth System in the Anthropocene.

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.

How motifs condition critical thresholds for tipping cascades in complex networks: Linking micro- to macro-scales.

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 Erdos-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 Erdos-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.

Armed-conflict risks enhanced by climate-related disasters in ethnically fractionalized countries.

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

Deep reinforcement learning in World-Earth system models to discover sustainable management strategies.

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.

Temporal organization of magnetospheric fluctuations unveiled by recurrence patterns in the Dst index.

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.

A perturbation-theoretic approach to Lagrangian flow networks.

Complex network approaches have been successfully applied for studying transport processes in complex systems ranging from road, railway, or airline infrastructures over industrial manufacturing to fluid dynamics. Here, we utilize a generic framework for describing the dynamics of geophysical flows such as ocean currents or atmospheric wind fields in terms of Lagrangian flow networks. In this approach, information on the passive advection of particles is transformed into a Markov chain based on transition probabilities of particles between the volume elements of a given partition of space for a fixed time step. We employ perturbation-theoretic methods to investigate the effects of modifications of transport processes in the underlying flow for three different problem classes: efficient absorption (corresponding to particle trapping or leaking), constant input of particles (with additional source terms modeling, e.g., localized contamination), and shifts of the steady state under probability mass conservation (as arising if the background flow is perturbed itself). Our results demonstrate that in all three cases, changes to the steady state solution can be analytically expressed in terms of the eigensystem of the unperturbed flow and the perturbation itself. These results are potentially relevant for developing more efficient strategies for coping with contaminations of fluid or gaseous media such as ocean and atmosphere by oil spills, radioactive substances, non-reactive chemicals, or volcanic aerosols.

Edge anisotropy and the geometric perspective on flow networks.

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.

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

We introduce the pyunicorn (Pythonic unified complex network and recurrence analysis toolbox) open source software package for applying and combining modern methods of data analysis and modeling from complex network theory and nonlinear time series analysis. pyunicorn is a fully object-oriented and easily parallelizable package written in the language Python. It allows for the construction of functional networks such as climate networks in climatology or functional brain networks in neuroscience representing the structure of statistical interrelationships in large data sets of time series and, subsequently, investigating this structure using advanced methods of complex network theory such as measures and models for spatial networks, networks of interacting networks, node-weighted statistics, or network surrogates. Additionally, pyunicorn provides insights into the nonlinear dynamics of complex systems as recorded in uni- and multivariate time series from a non-traditional perspective by means of recurrence quantification analysis, recurrence networks, visibility graphs, and construction of surrogate time series. The range of possible applications of the library is outlined, drawing on several examples mainly from the field of climatology.

Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution.

Potential paleoclimatic driving mechanisms acting on human evolution present an open problem of cross-disciplinary scientific interest. The analysis of paleoclimate archives encoding the environmental variability in East Africa during the past 5 Ma has triggered an ongoing debate about possible candidate processes and evolutionary mechanisms. In this work, we apply a nonlinear statistical technique, recurrence network analysis, to three distinct marine records of terrigenous dust flux. Our method enables us to identify three epochs with transitions between qualitatively different types of environmental variability in North and East Africa during the (i) Middle Pliocene (3.35-3.15 Ma B.P.), (ii) Early Pleistocene (2.25-1.6 Ma B.P.), and (iii) Middle Pleistocene (1.1-0.7 Ma B.P.). A deeper examination of these transition periods reveals potential climatic drivers, including (i) large-scale changes in ocean currents due to a spatial shift of the Indonesian throughflow in combination with an intensification of Northern Hemisphere glaciation, (ii) a global reorganization of the atmospheric Walker circulation induced in the tropical Pacific and Indian Ocean, and (iii) shifts in the dominating temporal variability pattern of glacial activity during the Middle Pleistocene, respectively. A reexamination of the available fossil record demonstrates statistically significant coincidences between the detected transition periods and major steps in hominin evolution. This result suggests that the observed shifts between more regular and more erratic environmental variability may have acted as a trigger for rapid change in the development of humankind in Africa.

Earth beyond six of nine planetary boundaries.

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.

Complex networks of interacting stochastic tipping elements: Cooperativity of phase separation in the large-system limit.

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.