Showing papers by "Stefan Thurner published in 2007"

Unanimity rule on networks.

Abstract: We present a model for innovation, evolution, and opinion dynamics whose spreading is dictated by a unanimity rule. The underlying structure is a directed network, the state of a node is either activated or inactivated. An inactivated node will change only if all of its incoming links come from nodes that are activated, while an activated node will remain activated forever. It is shown that a transition takes place depending on the initial condition of the problem. In particular, a critical number of initially activated nodes is necessary for the whole system to get activated in the long-time limit. The influence of the degree distribution of the nodes is naturally taken into account. For simple network topologies we solve the model analytically; the cases of random and small world are studied in detail. Applications for food-chain dynamics and viral marketing are discussed.

The window at the edge of chaos in a simple model of gene interaction networks

Abstract: As a model for gene and protein interactions we study a set for molecular catalytic reactions. The model is based on experimentally motivated interaction network topologies, and is designed to capture some key statistics of gene expression statistics. We impose a non-linearity to the system by a boundary condition which guarantees non-negative concentrations of chemical concentrations and study the system stability quantified by maximum Lyapunov exponents. We find that the non-negativity constraint leads to a drastic inflation of those regions in parameter space where the Lyapunov exponent exactly vanishes. We explain the finding as a self-organized critical phenomenon. The robustness of this finding with respect to different network topologies and the role of intrinsic molecular- and external noise is discussed. We argue that systems with inflated 'edges of chaos' could be much more easily favored by natural selection than systems where the Lyapunov exponent vanishes only on a parameter set of measure zero.

Towards a physics of evolution: Existence of gales of creative deconstruction in evolving technological networks

Abstract: Systems evolving according to the standard concept of biological or technological evolution are often described by catalytic evolution equations. We study the structure of these equations and find a deep relationship to classical thermodynamics. In particular we can demonstrate the existence of several distinct phases of evolutionary dynamics: a phase of fast growing diversity, one of stationary, finite diversity, and one of rapidly decaying diversity. While the first two phases have been subject to previous work, here we focus on the destructive aspects - in particular the phase diagram - of evolutionary dynamics. We further propose a dynamical model of diversity which captures spontaneous creation and destruction processes fully respecting the phase diagrams of evolutionary systems. The emergent timeseries show a Zipf law in the diversity dynamics, which is e.g. observable in actual economical data, e.g. in firm bankruptcy data. We believe the present model is a way to cast the famous qualitative picture of Schumpeterian economic evolution, into a quantifiable and testable framework.