Small world effects in evolution

Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study nonclustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small worlds that includes declustered networks and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.

Inducing coherence in networks of bistable maps by varying the interaction range

Ordinarily, two different topologies have been used to model spatiotemporal chaos and to study complexity in networks of maps: one where sites interact only with nearest neighbors (e.g., the standard diffusive coupling) and one where sites interact with all sites in the network (global coupling). Here we investigate intermediate regimes considering the interaction range as a free tunable parameter. The synchronization behavior normally seen in globally coupled maps is found to set in for interaction ranges considerably smaller than the system size. In addition, we analytically derive stability conditions for the onset of coherent states (full synchronization) from which the minimum interaction range needed to induce coherence in homogeneously coupled maps can be determined. Such conditions are also obtained for inhomogeneous situations when the coupling strength decreases linearly with the distance. The characteristic range for the onset of coherence is studied in detail as a function of model parameters.

Correlation effects in a simple model of a small-world network

We analyze the effect of correlations in a simple model of a small-world network by obtaining exact analytical expressions for the distribution of shortest paths in the network. We enter correlations into a simple model with a distinguished site, by taking the random connections to this site from an Ising distribution. Our method shows how the transfer-matrix technique can be used in the new context of small-world networks.