Statistical mechanics of complex networks

Complex systems, covering an impressive range of important applications from Systems Biology to Systemic Risk in the financial world, have experienced a tremendous increase of interest over the past decade. The essence of complex systems lies in the strong, long-range and often nonlinear interactions between their elements. The resulting collective (systemic) phenomena can not be understood by a superposition of individual and independent contributions. This makes an understanding of these systems exceedingly dificult. In this project, we propose to interpret the non-trivial correlation structure of complex systems as an abstract entity. These entities are the networks of interaction between the elements. Such networks - the `substrate of the correlation structure` of a system - can be naturally treated by means of statistical physics as dynamically rearranging sets of nodes and links, as has been impressively demonstrated recently. Networks, which eventually offer a promising more `holistic` approach to complex systems, can be studied independently from the details of the underlying systems. In this project, we plan to address several important problems associated with a statistical-mechanics formulation of networks. We focus on formal and physical aspects like extensivity violations, possibilities for equilibrium scenarios, entropy production, biologically or socio-economically motivated Hamiltonians and non-ergodicity, mainly. The project is motivated by the need from several branches of science for a better understanding (maybe even a unifying language) for complex systems. The project is intended to contribute to this goal.

Complex systems, covering an impressive range of important applications from Systems Biology to Systemic Risk in the financial world, have experienced a tremendous increase of interest over the past decade. The essence of complex systems lies in the strong, long-range and often nonlinear interactions between their elements. The resulting collective (systemic) phenomena can not be understood by a superposition of individual and independent contributions. This makes an understanding of these systems exceedingly dificult. In this project, we propose to interpret the non-trivial correlation structure of complex systems as an abstract entity. These entities are the networks of interaction between the elements. Such networks - the "substrate of the correlation structure" of a system - can be naturally treated by means of statistical physics as dynamically rearranging sets of nodes and links, as has been impressively demonstrated recently. Networks, which eventually offer a promising more "holistic" approach to complex systems, can be studied independently from the details of the underlying systems. In this project, we plan to address several important problems associated with a statistical-mechanics formulation of networks. We focus on formal and physical aspects like extensivity violations, possibilities for equilibrium scenarios, entropy production, biologically or socio-economically motivated Hamiltonians and non-ergodicity, mainly. The project is motivated by the need from several branches of science for a better understanding (maybe even a unifying language) for complex systems. The project is intended to contribute to this goal.