Systemic risks pertaining to systems of nature, technology, and society often manifest themselves as dynamic macroscopic phenomena that result from actions of agents, among each other and with their environment, on the microlevel in a complex system. In this context, we use the notion of agents in a most general perspective, ranging from atoms and molecules in physical and chemical systems up to humans in socioeconomic systems. Clearly, the possible interactions between such different agents vary widely, from simple and well-understood physicochemical interaction laws between molecules up to nonlocal, adaptive decisions and interactions including memory effects within and among human individuals. It goes without saying that such widely different interaction patterns preclude any general predictive theory for the system’s dynamic behavior and the associated systemic risks. Inherent properties of complexity limit their predictability and make them often counter-intuitive. Yet, and this is the primary message of this communication, some very fundamental patterns of dynamic behavior do not depend crucially on details of the agent’s interactions with the remarkable effect that systems, with very different types of agents, can show in terms of rather similar patterns of dynamic behavior. We refer to this phenomenon as homomorphism. This insight, together with using the instruments of complexity science, will be shown to support significantly the understanding and governance of systemic risks.
Complexity has many aspects and this is reflected by the many different branches and instruments of complexity science. In the context of systemic risks, we make use in particular of those typically associated with dynamic structure generation in open systems under strong external and/or internal perturbations. We refer to such situations as nonequilibrium with reference to the notion of equilibrium typically used in physical chemistry: that is, a state without any change of its macroscopic behavior over time in spite of rapid and active dynamic processes on the microlevel. They are characterized by tipping and cascading phenomena along with dynamic pattern formation. We use the term nonequilibrium complexity. Such processes have been studied in detail and depth in the natural sciences, notably physics and chemistry. Interestingly and most importantly, as already mentioned above, it can be shown that the principal patterns of behavior revealed there can be found across all domains, from biology to ecology up to socioeconomics. Thus, for the benefit of analyzing systemic risks in any domain, advantage can be taken of the detailed knowledge accumulated by studies of relatively simple systems in physics and chemistry, studies that would not be possible in more complex systems such as those of biology, ecology, or society. In the following paragraphs, we summarize some of these insights of nonequilibrium complexity relevant to systemic risks with the emphasis on generating awareness of their provenience and the benefit derived from recognizing this.
Some Common Features of Systemic Risks
A collection of empirical evidence associated with systemic risks in the framework of nonequilibrium complexity science reveals a fundamental homomorphism with respect to their basic structures across all domains, based essentially on universal strategies of information processing between the agents and the global system. This opens up a large body of empirical knowledge that forms the basis for their understanding and analysis.
Agents and Emergence
Dynamic structures, associated with systemic risks, are phenomena of emergence, typically appearing out of system instability. They are collective effects resulting from the elementary actions of the agents on the microlevel. So, they are only observable on the macroscopic level, not on the microlevel of the agents. In physicochemical systems, single molecules do not display the coherent patterns observable in flow or chemical reaction systems with a large number of molecules; in social systems single individuals do not show the patterns of mass phenomena and political mass radicalism frequently observed in situations with many human individuals. Identifying the microlevel of the agents for analysis of dynamic structure generation is a matter of intelligent choice. Any system will display a more or less complicated stratification over various levels of interactions, and so it turns out that the adequate choice of the microlevel is a matter of the theoretical or empirical information that is available. To illustrate, it is instructive to look at one of the most elementary and least stratified model system of dynamic structure generation in physics—the laser. The agents to consider are the atoms of the laser material and the photons of the light wave (Haken 1977). It is the nonlinear and originally localized feedback interaction between the atoms of the laser material and the photons of the light field that is responsible for the emergence processes stabilizing the macroscopic light wave. Information processing of the agents is affected by local statistical sampling from patterns that are able to spread over the whole system. The interactions between these agents are known from quantum-electrodynamics and so the analysis can be based on this theory, and thus a full understanding of the structure generation mechanisms can be obtained. The importance of this analysis goes much beyond this particular physical device. It constitutes a metaphorical interpretation of these effects as a learning and adapting behavior of the agents in principle. The laser represents a model system that is rather informative and instructive for dynamic structure generation in general and for associated systemic risks in a wide variety of domains.
For autocatalytic chemical systems, a typical model system for dynamic structure generation in chemistry, such elementary information is not available in the general case. The analysis of structure generation is thus rather based on the mesolevel of empirical gross reactions for selected molecular species and on their time evolution in terms of associated rate equations. Strong nonlinear effects as provided by the autocatalytic reactions with the associated feedback effects are a necessary condition for structure generation in these systems (Prigogine and Lefever 1968). By mechanisms in principle analogous to those in a laser, initial and localized patterns such as stripes and rings of alternating color spread over the whole system by information processing, and lead to a global pattern in the system.
It is rewarding to apply these fundamental insights about the role of agents and their particular types of interactions in dynamic structure generation, which are gained by analyzing relatively simple physical and chemical systems, to the much more complicated socioeconomic systems. Here, agents may be organizations, human individuals, social groups, and at whatever level for which proper information is available or sensible modeling can be applied. Besides structure generation by external control, such as in the systems of physics and chemistry addressed above, there are many examples of self-organized emergence in social systems even without external impact. Markets can be seen as emerging from the individual actions of traders, business organizations as emerging from the activities of their owners and employees, in addition to the actions of groups such as legislators, lawyers, advertisers, and suppliers. The choice of the agents is not separable from the definition of the interactions between them and their environment. Although the agent’s interactions are clearly entirely different from those physicochemical systems mentioned above, basic mechanisms of dynamic structure generation and pattern formation are pretty much analogous. In socioeconomic systems, similar to the physicochemical model systems, strong and reinforcing interactions generate feedback loops and circular causality between the macroscopic structures and the actions of the agents. To paraphrase Clifford Geertz (1973, p. 5), we are animals suspended in webs of meaning that we ourselves have spun. Agents learn from and adapt to the environment created by themselves. It is frequently the interaction of human individuals as agents with a field of information derived from the media, public opinion, or other sources that stabilizes the structures in socioeconomic systems. In this way, local patterns are again able to spread over the whole system by appropriate information distribution.
Tipping and Cascading
The object of study in systemic risks is invariably a system along with its boundaries. Boundary conditions may keep the system in a stable macroscopic state with continuous microdynamic change. But when the boundary conditions exceed threshold values they can drive the system into a regime of instability out of which new dynamic structures may suddenly emerge when appropriate internal conditions prevail. This phenomenon is referred to as tipping or as a bifurcation.
In nonequilibrium thermodynamics as the appropriate theory of molecular systems, it is shown that in such situations, unlike situations in equilibrium or close to it, driving forces and resulting effects have a nonlinear relationship to each other. A potential function then no longer exists (Prigogine 1980). As a consequence, small and random fluctuations that are normally inconsequential may trigger a dramatic change in behavior, referred to as a phase transition. Molecular interactions like information processing in a network spread the new state in a cascading process over the whole system. A well-known example from chemistry is the local emergence and subsequent spatial cascading of a dynamical chemical structure, which creates a regular chemical space-time pattern (Gray and Scott 1990). Similar patterns are found in the study of pandemics and the spread of disease (Chen et al. 2017). Further examples are failures of infrastructure, such as the electrical grid (Koonce et al. 2008), the breakdown of a financial system (Hurd et al. 2016), or such phenomena as public opinion formation or economic innovations (Weidlich 2000).
Phenomenologically, an approach to such a tipping point becomes noticeable by an amplification of fluctuations. These fluctuations are of a random nature, which makes the future of the system unpredictable in detail, although the system normally can choose only out of a few macroscopic patterns. As a consequence of nonlinearity, the average dynamical behavior of the system in such situations no longer corresponds to the average of the individual activities of its agents. Rather, a new structure may originate from the amplification of one particular microscopic fluctuation at the right moment and under the right conditions. A momentary individual activity can thus determine the dynamic future of the macroscopic system. These effects make intuitive predictions of system behavior rather questionable. Tipping and cascading phenomena can be and have been studied in detail in some prototype systems of physics and chemistry based on rigorous theory. In more complex domains, for example, socioeconomics, we observe essentially the same phenomena, a homomorphism, although a detailed theoretical analysis is still missing. But recognizing the analogies paves the way for an adequate ordering of the empirical observations during dynamic pattern formation across domains.
Parameters Indicating Instability
An immediate, practically relevant consequence of transferring established knowledge from physics and chemistry to more complex domains is the recognition that parameters exist that indicate instability even in those domains where there is no theory to directly provide them. As theoretically established in physics and chemistry, the dynamic structure generation in such unstable regimes is initiated by selection processes on the microlevel. In the course of elementary fluctuations, the system tests its various modes of dynamic behavior, which may be considered as a population of macroscopic patterns. Hierarchies of time scales and strengths of interaction during the elementary dynamics are responsible for a selection process that stabilizes the new dynamic structures. The same principles identified by rigorous theory in a laser, as well as in flow or chemical pattern formation, are effective in other domains.
In socioeconomic systems, such effects lead to the formation of a stratification and topology of the system with widely autonomous and interacting substructures such as families, organizations, ethnic subgroups, and the like. As a result of these selection processes only few modes of behavior win the competition and become visible macroscopically. They can be described in terms of so-called order parameters, that is, the relevant macroscopic variables for representing the dynamics of a system. This results in a drastic reduction of complexity of the dynamics close to tipping points as compared to the chaos on the microlevel. In this sense, we witness the emergence of order out of chaos (Prigogine and Stengers 1984).
It is therefore promising to look for simple macroscopic parameters that describe the approach to instabilities. Generally, such parameters are nondimensional in nature, and balance enhancing and hindering effects by a suitable combination of external and internal quantities. Examples from physics and chemistry are the Reynolds-number or Rayleigh-number in flow systems and the ratio of pumping energy to light loss in a laser, where these parameters are easily derived from theory. Similar parameters have been identified empirically in socioeconomic systems, such as the ratio of local uprisings to police interventions in the forefront of revolutions (Schröter et al. 2014), the size of the economy to the amount of private debt in the onset of a financial crisis (Minsky 2008), or the index of conflict-related news before the outbreak of war (Chadefaux 2014). In cascading processes, the relation of time scales for propagation and adaption are the crucial parameters that announce instability such as that of chemical reaction to diffusion rates in chemical pattern formation. The elementary selection processes on the microlevel are ultimately responsible for a hierarchy in macroscopic time scales. For example, the universally observable phenomenon of a slow approach to an instability regime is followed by a sudden phase transition with systemic risks in widely differing domains. Typical examples are the sudden tipping phenomena of ecosystems or social upheavals after a long time of enduring stress.
The dynamics of the system has a past and a future about which some general statements can be made. The approach to instability is frequently followed by a phase transition in the form of a bifurcation, that is, a sudden change of system behavior without external design. Even in the particularly simple flow system of the Bénard convection, two different structures, for example left and right turning convection cells, become available at a phase transition from the unstructured state close to equilibrium (Bénard 1900). The system then chooses at random one possible stable branch of the dynamics and thereafter proceeds in a deterministic evolution until a new regime of instability is reached. Here again a phase transition with the system making a choice may take place. The overall dynamics is thus determined by the fact that, depending on increasing temperature difference as impact over the boundaries in the Bénard convection, the system chooses stable states by sudden phenomena of self-organization, with deterministic and smooth periods of development in between, and thereby follows a particular path. The behavior of the system thus exhibits an individual history, consisting of an interplay between chance and determinism, and the final flow structure may be quite complicated.
Analogous patterns are visible during structure generation in socioeconomic systems, be it traffic congestion, political revolution, technological innovation, or urban settlements. For example, the adoption of one of a pair of alternative technologies within a society or the market success of a particular company can be greatly influenced by minor contingencies about who chooses which technology or which company at an early stage. This pattern clearly is reminiscent of the butterfly effect usually observed in chaotic systems. This early choice can determine the further fate of the system and explains the remarkable success of one technology or one company over others when the winner takes all. Such a path dependence excludes any purely local and momentary origins of systemic risks, for instance of a stock market crash (Sornette 2003) or the recent refugee crisis (Lucas 2016). It is indispensable to study the history of the system if one aims at an adequate understanding of the dynamics of the system. Rather advanced mathematical methods of time series data analysis are available to discover early warning signals.
Eventually, the system will be drawn to a particular attractor as its ultimate dynamic domain. This end result may be simply a fixed state, but it may also be of a more complicated nature, such as periodic oscillation between fixed states or even chaos, depending on the system’s parameters and history. If more than one attractor is present each one will be approached by the dynamics of the system from its individual basin of attraction as the initial condition, such as a water divide establishes two basins of attraction deciding to which side the rainwater will flow. It is crucial to analyze attractors, since they are the basis for suitable mitigation measures of systemic risks.
As shown above, there is strong empirical evidence of homomorphism with respect to fundamental dynamic patterns associated with systemic risks. Yet, more support and instruments for practical exploitation on the basis of complexity science can be provided by turning to quantitative modeling.
This homomorphism is mirrored in the mathematical theory of dynamic systems. It is shown there that a class of rather different mathematical models tailored to rather different systems essentially reproduces the same patterns of dynamic behavior, notably universal types of attractors. Contrary to many standard problems of physics and chemistry, an individual mathematical model in complexity theory is just the beginning of an understanding; its evaluation over time, be it in the form of an iterated function or a differential equation, produces unforeseen dynamical structures. Well- defined rules, often surprisingly simple, when applied in active repetition without any intentionality over and over again, lead to an evolution in time showing a remarkable creativity and richness of structures. This result is not to be expected from the simple underlying model specifying the rules. These structures in terms of the associated attractors can be visualized in so-called bifurcation diagrams that are specific in detail but universal in the basic patterns associated with the dynamical behavior of complex systems, up to amazing universalities even in the range of chaos.
The time behavior of a complex system, as described by such deterministic equations, is not entirely unpredictable. As different as the models may be for one system or another, the attractors can be predicted quite well. Although in a case where there is more than one attractor, the particular choice the system takes will be stochastic and thus unpredictable. Even in the case of chaos, the attractor of the system, referred to as a strange attractor and representing the general framework of long-term system behavior, is predictable from a model, even though the particular choice of the dynamics along this attractor over time is sensibly dependent on the initial conditions and thus clearly unpredictable over sufficiently long times. A prediction of short run evolution is possible even in a chaotic system, contrary to any long-time development, as is well known from domains such as weather forecasting.
More frequently than not a mathematical model is not available. But if data over time can be acquired, these can be analyzed by appropriate mathematical methods to extract patterns of dynamic behavior that, in favorable cases, will allow some prediction to be made about the future behavior of the system. This predictive ability includes the occurrence of systemic risks. Examples are known from challenges as different as stock market analyses, weather forecasts, heart attacks warnings, and many other cases.
More specific information about the rules governing the emergence of dynamic structures and systemic risks can be introduced in quantitative models through methods of computer simulation. In such simulations the rules on the microlevel can be tailored to the information available about a system. In particular, the stochastic nature of such data, often responsible for the empirically observed unpredictability, notably in socioeconomic systems, may be taken into consideration.
In physical chemistry, the emerging system state—the relationship between the microlevel, the molecules, and the macrolevel—is quantitatively accessible through a particular type of computer simulations within the framework of statistical mechanics (Lucas 2007). In molecular dynamics the dynamics of molecules is studied on the basis of Newton’s equations of motion and appropriate interaction models. The equilibrium attractors of dynamical properties are found in the limit of long runs by time averaging. In a pioneering paper, Alder and Wainwright (1957) demonstrated that a simulation of the dynamics of a relatively small number of molecules, modeled crudely as hard spheres, explained the emergence of solid order in a fluid system of randomly distributed molecules under suitable external and internal conditions. Although no condensation phenomena could be found in a hard sphere system, it was shown later that such a phase transition appeared as soon as attractive forces were added to the model of intermolecular interactions. When a realistic model of the interactions, as accessible today from quantum mechanics, was introduced, predictions in good agreement with experiments could be achieved for the equilibrium behavior of fluid systems (Luckas and Lucas 1989). These early results in physical chemistry indicate the power of such simulation approaches. They draw attention to the phenomenon common to complex systems that basic properties of the system’s dynamics can be studied by rather crude models of the interaction rules of the agents.
In more general systems, notably those of socioeconomics, analogous types of quantitative modeling have been established. The phenomena associated with systemic risks invariably emerge from agent actions that can be studied and modeled on various levels of depth and empirical trustworthiness, depending on the system under study. As a consequence, there is, beyond qualitative pattern recognition and understanding, a quantitative access to their analysis by introducing these models into computer simulations or mathematical equations within the framework of complexity science. Various approaches in this category have been developed. Most prominent are agent-based computer simulation (Railsback and Grimm 2011) and mathematical formulations in terms of master equations (Weidlich 2000). These are similar in spirit, although differing in detail, to computer simulation of molecular systems, and much has been learnt from these roots.
In multiagent simulation formalisms, based on the direct pair and higher order interactions between agents and their environment, it can be demonstrated (as found in the physicochemical systems) that rather simple elementary rules for the agents may result in the emergence of a rather complex dynamic behavior of the system as a whole. But the agents in such systems differ fundamentally from molecules. They have many more internal properties, that is modes of potential behavior, and the excitation of these degrees of freedom depends on the situation of all the subsystems. Care must be taken to account properly for the modes of possible actions in a given system and to model the actions properly, based on empirical analysis. In particular, the actions are not only direct and instantaneous as in molecular systems but they also may be intermediate, antisymmetric, and associated with subjective wishes and memories. In physicochemical systems, wholeness patterns simply emerge from the undirected and short-range interactions of atoms and molecules; in socioeconomic systems, however, patterns emerge by intentional activities of human individuals in an environment that humans themselves create. By bottom-up effects the agents’ actions in the form of cultural and economic activities generate a collective field. This is a sociocultural field that acts back top-down upon the agents in an ordering manner by influencing their actions. Stochastic effects that reflect the uncertainties of the actual agent behavior may, among other strategies, be included by adding random noise to their action rules.
Clearly, any tight analogy to physics and chemistry systems is here restricted to the method of analysis and the emergence of the typical phenomena associated with dynamic systems discussed earlier. In detail, of course, there is a richness of behavior in socioeconomic systems far beyond what can be found in the relatively simple systems in physics and chemistry. The analogies revealed through insights form the physical world can, first, act as a heuristic tool to look for similar phenomena in the social world, and, second, to provide the basis for exploring functional or even causal properties that may lead to further implications that are eligible for statistical analysis of data sets.
An illustration of agent-based computer simulation studies that is still close to molecular dynamics in physical chemistry is the analysis of crowd disasters during mass events (Moussaid et al. 2011). The goal of such studies is to clarify the cause of such disasters and then to look for governance strategies to improve crowd safety. The dynamics of crowd behavior is complex and often counterintuitive. A systemic failure is usually not the result of one single event on the microlevel of the individuals. Instead, it is the interaction between many events that cause a situation to get out of control. Computer simulations are able to shed light on the crucial and generalizable phenomena and behavioral patterns of human agents responsible for the emergence of disasters in mass events. Crowded situations, where instinctive physical interactions dominate over intentional movements, give rise to global breakdowns of coordination, and cause strongly fluctuating and uncontrollable patterns of motion to occur (Helbing et al. 2015). Whereas simple proportionalities exist between crowd density and crowd flow in uncritical situations, this assumption fails at crowd densities passing a critical threshold. In such cases, fatalities may result from a phenomenon called crowd turbulence, notably close to geometrically sensitive situations (Hoogendoorn and Daamen 2005). The occurrence of crowd turbulence can be understood and reproduced in computer simulations by means of force-based models in combination with Newton’s equations of motion to describe the microlevel dynamics. The most important lesson learned with respect to governance is to stay away from the threshold of local density. The problem is that local densities are hard to measure without a proper monitoring system, which is not available at many mass events. Existing technologies, based on video, WiFi, GPS, and mobile tracking, do offer useful monitoring systems that allow for real-time measurements of density levels and predictions of future crowd movements. By a combination of computer simulations, real time monitoring systems, and complexity science, important progress in the governance of systemic risks during mass events has been achieved (Wirz et al. 2013; Ferscha 2016).
In the master equation formalism, the microlevel dynamics being defined by considerations, decisions, and actions of individuals is modeled by probabilistic elementary steps in which the individuals perform specific actions, thereby changing their properties and thus the macroconfiguration as a whole, referred to as the socio-configuration. Particular transitions may or may not happen, guided by decision and action generating motivations, which are quantified by a priori undetermined parameters. While this is analogous to the procedure in agent-based simulation, different use is made of the elementary steps. Balancing in- and outgoing probabilities leads to the master equation, a differential equation for the probability function of the social configuration. Long time attractors are found by suitable mean value operations.
As an illustrative example of the master equation formalism, we consider dynamic structure generation in the migration of interacting populations (Weidlich and Haag 1980). Modeling this dynamic sheds light on and supports understanding of the emergence and evolution of parallel societies as a major systemic risk in modern societies. The agents in the migration phenomenon are the n individuals of the system. This may be a country or a city with C different and distinguishable areas i, such as regions of a country or quarters of a city. The analytical goal is to scrutinize the conditions under which an original social configuration, for example, a relatively homogeneous distribution of individuals over all areas in the system, may become unstable and perform a transition to a nonhomogeneous distribution, such as the stable formation of ghettos or the rhythmic oscillation of otherwise inhomogeneous distributions. Migration of interacting populations and its effect on the population distribution in a system is the result of economic, social, and in particular multicultural interactions between the individuals. External impacts may be political regulations, which set forth rules for the living together of individuals in the society, or also the informational flow into the system from other societies, where migration phenomena have been demonstrated to be either advantageous or harmful to those individuals who have decided to migrate. Internal effects may be the advent of strong cultural or ethnic feelings leading to agglomeration or segregation movements. A critical combination of external and internal random fluctuations in a selection process may drive the system out of one societal configuration into another one. An elementary action is the transition of one individual from region i to region j, which changes the social configuration in a particular way. The associated transition probability is formulated in terms of the attractiveness of such a transition as felt by the considered individual. In the absence of any deterministic theory for the dynamics on the elementary level, the transition probability has to be formulated empirically, but so doing must make use of the insights that nonequilibrium complexity provides. Solving the master equation shows that starting from a homogeneous distribution and turning on external and internal impacts in terms of the appropriate nondimensional parameters, the system approaches a particular regime of instability from where a sudden phase transition to a particular new distribution may occur on exceeding a threshold. Out of this distribution further thresholds that induce additional phase transitions to changing distributions will appear. Over long periods, the process of dynamic structure generation will yield to an attractor, calculable from the master equation, which may be a homogeneous distribution of the individuals over the areas, a stable ghetto structure, or also an oscillation between these states.
Many studies of computer simulations in the social sciences are available in the literature. Some key references with extensive further citations include Weidlich (2000), Gilbert (2007), and Helbing (2015). These works are rooted in the established methods of such studies in physics and chemistry, and reveal analogous or even homomorphic fundamental patterns of behavior by appropriate applications of statistical methods. But, in detail, they address a much more comprehensive and complex richness of patterns, which makes those patterns much more difficult to interpret. Since fundamental interaction laws, contrary to the systems of physics and chemistry, are not available from independent theories, they must be parametrized. As a consequence, the final outcome of such studies is, again contrary to the systems of physics and chemistry, not a prediction of the future behavior of the system but rather a study of scenarios that describe what may happen under particular conditions. These scenarios are not only thought experiments of what one could imagine, but rather constitute consistent and coherent simulations of further developments of complex and dynamic systems. These possible visions occur within a range of potential “futures” that depend on collective human actions as well as unforeseen changes in context conditions.