Time scale separation and heterogeneous off-equilibrium dynamics in spin models over random graphs

We study analytically and numerically the statics and the off-equilibrium dynamics of spin models over finitely connected random graphs. We identify a threshold value for the connectivity beyond which the loop structure of the graph becomes thermodynamically relevant. Glauber dynamics simulations show that this loop structure is responsible for the onset of dynamical features of a local character (dynamical heterogeneities and spontaneous time scale separation), consistently with previous (experimental and numerical) studies of glasses and spin glasses in their approach to the low temperature phase.Comment: 5 pages, latex, 2 postscript figure

Maximum flow and topological structure of complex networks

The problem of sending the maximum amount of flow $q$ between two arbitrary nodes $s$ and $t$ of complex networks along links with unit capacity is studied, which is equivalent to determining the number of link-disjoint paths between $s$ and $t$. The average of $q$ over all node pairs with smaller degree $k_{\rm min}$ is $_{k_{\rm min}} \simeq c k_{\rm min}$ for large $k_{\rm min}$ with $c$ a constant implying that the statistics of $q$ is related to the degree distribution of the network. The disjoint paths between hub nodes are found to be distributed among the links belonging to the same edge-biconnected component, and $q$ can be estimated by the number of pairs of edge-biconnected links incident to the start and terminal node. The relative size of the giant edge-biconnected component of a network approximates to the coefficient $c$. The applicability of our results to real world networks is tested for the Internet at the autonomous system level.Comment: 7 pages, 4 figure