Critical transitions and systemic risk in dynamical networks

Financial markets and more general socio-economic systems can be extremely fragile. The recent history of financial market crashes is well-documented and their impact on society at large has been very significant. In response to these crises, classical available tools of mathematical finance seemed to be of very limited utility, e.g., the president of the European Central Bank (ECB) Jean Claude Trichet remarked in 2010: ``As a policy-maker during the crisis, I found the available models of limited help. In fact, I would go further: in the face of the crisis, we felt abandoned by conventional tools.`` Clearly, this is a direct call to develop additional fundamental research methods. In the project we aim to take steps to develop these additional tools at the interface between applied mathematics, theoretical physics, complex systems and economics. Our focus is to study models based upon network science, where economic players are represented as nodes and their interactions as links. Clearly, these networks have to be `dynamic` in many respects, e.g., the current status or strategy of players may change and the links of economic interactions are not static. Although constructing simple model network systems is comparatively easy, their analysis and prediction turns out to be extremely difficult. Therefore, it is beneficial to focus on the network dynamics for certain cases. In the project we focus on behavior of dynamical networks near instabilities, i.e., when there is a risk that the system loses its regular function. Recent studies in different sciences as well as new mathematical theories suggest that in very small networks, one may be able to detect that a system is near a critical threshold based upon measuring its fluctuations. It is one main goal of the project to try to extend this idea of early- warning signs to complex network dynamics. Of course, this leads to fundamental mathematical questions, e.g., how do we relate small to large models or which variables can be used in large models for prediction? Furthermore, modeling and large-scale numerical simulations for socio-economic systems and application to real data sets of inter-banking networks will allow us to validate our theoretical findings in more concrete situations. The main goal of the project is to develop and prove the correctness of mathematical methods and to investigate the relation to networks subject to systemic risk in socio- economic toy-models. This will bring us, if only a bit, closer to prevent and mitigate systemic risk based upon solid foundations in applied mathematics and complex systems.

We studied early-warning signs for collapses in adaptive network dynamical systems. In such systems the state of the nodes co-evolves with the interaction structure. We showed that a sizable class of such model systems exhibits a quantization in the state of their nodes before they collapse. We showed that this is linked to the presence of one last remaining directed cycle in the network. Since the number of such feedback cycles often indicates the stability of the system, we therefore found a systemic early-warning sign. Moreover, the quantization manifests itself only in the relative states of the nodes without reference to the underlying network, making it applicable to cases where the network structure is not directly accessible. This class contains all those models in which the dynamics behaves effectively or approximately linear, as it is the case in catalytic networks or close to an equilibrium. In particular, the Jain-Krishna model of species evolution falls into this class, so that we could provide a very reliable early-warning sign for this model. We also studied early-warning signs for a second class of adaptive network systems in which the system slowly moves towards a tipping point. We considered a classical representative of this class, an adaptive epidemic spreading model in which infected individuals may recover or spread the contagion to a susceptible neighbor, but susceptible individuals may detach and re-wire themselves from infected neighbors. Here we showed both analytically and by means of simulations that the onset of the critical transition shows a maximum in certain network-derived quantities. We focused here on the transition from an epidemic state into a situation where the epidemics dies out due to sufficient adaptive re-wiring. We then also found that the classical theory of early-warning signs can completely fail on this class of system. The classical theory for such systems says that the critical transition is preceded by an increase in the variability in a power-law fashion that allows one to extrapolate the critical point at which the transition occurs. However, applying this technique to the adaptive epidemics may lead to vastly wrong predictions, as we showed, since some of the network-based quantities do not pick up the signature of the critical transition, but instead behave as though the transition was at a different point. We could show this again analytically by means of approximations and numerically by means of simulations. With these insights into the early-warning theory of two classes of adaptive network systems we then reached out to the community via conference talks and publications and deepened our understanding by modifications. Through ongoing projects we continue our contributions to this field of research.