Enhanced Flow in Small-World Networks

Abstract

The proper addition of shortcuts to a regular substrate can lead to the formation of a complex network with a highly efficient structure for navigation [J. M. Kleinberg, Nature 406, 845 (2000)]. Here we show that enhanced flow properties can also be observed in these small-world topologies. Precisely, our model is a network built from an underlying regular lattice over which long-range connections are randomly added according to the probability, Pij ∼ r−α ij , where rij is the Manhattan distance between nodes i and j, and the exponent α is a controlling parameter. The mean two-point global conductance of the system is computed by considering that each link has a local conductance given by gij ∝ r−C ij , where C determines the extent of the geographical limitations (costs) on the long-range connections. Our results show that the best flow conditions are obtained for C = 0 with α = 0, while for C ≫ 1 the overall conductance always increases with α. For C ≈ 1, α = d becomes the optimal exponent, where d is the topological dimension of the substrate. Interestingly, this exponent is identical to the one obtained for optimal navigation in small-world networks using decentralized algorithms.