A minimal model for congestion phenomena on complex networks

We study a minimal model of traffic flows in complex networks, simple enough to get analytical results, but with a very rich phenomenology, presenting continuous, discontinuous as well as hybrid phase transitions between a free-flow phase and a congested phase, critical points and different scaling behaviors in the system size. It consists of random walkers on a queueing network with one-range repulsion, where particles can be destroyed only if they can move. We focus on the dependence on the topology as well as on the level of traffic control. We are able to obtain transition curves and phase diagrams at analytical level for the ensemble of uncorrelated networks and numerically for single instances. We find that traffic control improves global performance, enlarging the free-flow region in parameter space only in heterogeneous networks. Traffic control introduces non-linear effects and, beyond a critical strength, may trigger the appearance of a congested phase in a discontinuous manner. The model also reproduces the cross-over in the scaling of traffic fluctuations empirically observed in the Internet, and moreover, a conserved version can reproduce qualitatively some stylized facts of traffic in transportation networks

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA