Statistical Mechanics
Traditional science usually fails when confronted with complex systems. For example the classical view of statistical physics and thermodynamics asserts non-applicability to complex systems. We think that this is not the end of a development, but that paradigms and tools from these fields can be utilized for a fundamental understanding of complex systems.
We revisit the question of the nature of ‘information’ in interacting or correlated systems, where the traditional Boltzmann-Gibbs-Shannon entropy loses its meaning. We constructively design an entropy from thermodynamical principles which works for correlated systems and show that a ‘thermodynamics for CSs’ is perfectly reasonable.
Our approach is highly universal and incorporates classical simple systems (Boltzmann-Gibbs systems) and those characterized by power-laws (Tsallis systems) as special cases. Our approach extends to practically all characteristic distribution functions observed in nature. Our research activities include work on:
- Entropies for complex systems
- Thermodynamics of complex systems
- Stability of complex systems
- ‘Superstatistics’
Network Theory
Many CS are composed of elements whose interactions are given by networks. Examples include: genetic networks, metabolic networks, social communication networks, trading networks, banking networks, the WWW, and production networks in economies, etc. The structure of networks is often related to their function. We empirically study the structure (topology) of real-world networks.
Many processes do not take place in space and time, but on Many processes do not take place in space and time, but on networks and time – the classical view of space is reduced to the topologies of networks. We study and model dynamical processes on networks and derive consequences for efficiency, stability, adaptability, etc. Examples include genetic networks, credit networks, or networks of financial flows within an economy.
Many networks are not static but evolve over time. Often the characteristics of the nodes of a network (states) influence the dynamics of re-linking and consequently the emergent network properties. We design and study models where networks influence states of nodes, and where states of nodes influence network connectivity. These co-evolving networks are used e.g., for voter dynamics, strategic marketing, communication networks, or opinion spreading models.
The structure of a network determines much of its performance/efficiency, the function determines much of the structure. We try to understand these mutual influences and derive practical consequences. We develop a statistical mechanics of and for networks.
Systemic Risk
As the recent financial crisis of 2008–2010 has shown, we do neither understand nor know how to deal with systemic risk, i.e. the risk of collapse for large-scale complex systems. We try to understand the preconditions under which collapse of complex systems becomes possible and likely. Methods employed here involve network theory, synchronization dynamics on networks, combinatorics, agent-based models, game theory and non-linear dynamics.
For evolution systems we were able to show that they all have critical points at which the probability of systemic collapse approaches one, i.e. collapse becomes almost certain. In other words, evolution systems contain the seed for their collapse within themselves. As an alternative to management of systemic risk, we propose to make efforts to identify these critical points within real world complex systems, such as ecological, financial and social systems.
Physics of Evolution
We aim at a consistent mathematical formulation of evolution systems as systems with co-evolving boundary conditions. Such systems are not treatable with traditional mathematics. We deviate from classical Darwinian thinking by interpreting fitness landscapes as co-evolving structures. We were able to show that evolution systems have a phase structure, i.e. that one and the same system can exist in different modes or phases: they can be in a mode of ever-increasing diversity, or in a dull mode of almost stationary levels of low diversity. Transitions from one phase to another are associated with booms or crashes in diversity.
Implications of this work are not limited to biological evolution, but apply to technological innovation, economics, theoretical chemistry and many other processes in the living, social and physical world. In particular we make efforts to formulate economy in an evolutionary framework. With this we are able to understand and quantify key elements of Schumpeterian economics, such as gales of destruction and creative destruction.