Groupe de Physique Statistique/ Arbeitsgruppe Statistische Physik

Empirical studies of many complex networks ranging from social networks to power grids and the Internet have revealed that in many cases properties like the node degree are power law distributed e.g. p(k)~ k^-l. This implies that the role of the nodes e.g. with respect to the connectivity differ considerably. A number of evolutionary growth schemes have been proposed that may reproduce these properties and explain the often non-equilibrium statistics. Our focus is on percolation phenomena in such networks. Here, percolation is defined by the birth of a giant connected component (incipient cluster). For a degree distribution p(k) as above the percolation critical exponents will depend on l. For a class of treelike networks we explicitly derive the transport properties of these networks and find the full density of states as well as scaling relations for the dynamical scaling exponents [1]. As a specific example of complex networks we analyze the public transport (PT) networks of a number of major cities of the world. While the primary network topology is defined by a set of routes each servicing an ordered series of given stations, a number of different neighborhood relations may be defined both for the routes and the stations. E.g. one either defines two stations as neighbors whenever they are serviced by a common route or only if one station is the successor of the other in the series serviced by this route. Previous studies of PT have mostly been restricted to much smaller networks and did not observe power law behavior for which we find clear indications in the larger of the networks that we analyze [2,3]. Removing nodes from the network we define paths to percolation. The corresponding behavior strongly depends on the path chosen, i.e. proceeding either by random removal or targeted attack of nodes with high centrality. Our findings for the relation between the topology and vulnerability of these networks is supported by simulations of a model for the evolution of PT networks that we propose. [1] F. Jasch, CvF, A. Blumen. PRE 68:051106 (2003). [2] CvF, Yu. Holovatch, V. Palchykov Condens. Matter Phys. 8:225 (2005). [3] CvF, T. Holovatch, Yu. Holovatch, V. Palchykov (2006, in preparation)