# Dresden 2006 – wissenschaftliches Programm

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# DY: Dynamik und Statistische Physik

## DY 10: Statistical Physics of Complex Networks I

### DY 10.3: Vortrag

### Montag, 27. März 2006, 10:15–10:30, H\"UL 186

**Emergence of Hierarchical Structures in a Stochastic Network Model** — •Michael Koenig, Stefano Battiston, and Frank Schweitzer — Chair of Systems Design,ETH Zurich,CH-8092 Zurich, Switzerland

We investigate a network model governed by processes on two different time scales: The short time scale describes the eigendynamics of the nodes, a feature often neglected in network models. The long time scale describes the change of the network structure itself which represents the interactions between the nodes. Each node is characterized by a scalar variable, representing for example “size” or “output”, in a stochastic equation with auto-catalytic and hetero-catalytic growth terms. For the dynamics of the network, we consider different sets of rules for rewiring the links according to the output of the nodes. For example, a rewiring of any link between two nodes is accepted iff this increases the output of both nodes. Starting from a random graph, the dynamics leads to a saturated state characterized by an optimized output of the system (Nash equilibrium). We find that this equilibrium structure corresponds to a hierarchy in the output distribution. Averaging over different network realizations, we further obtain power-law like behavior for other network variables, such as the distribution of links, clustering coefficients and the number and length of cycles in the network.