Enhanced Flow in Small-World Networks

The small-world property is known to have a profound effect on the navigation efficiency of complex networks [J. M. Kleinberg, Nature 406, 845 (2000)]. Accordingly, the proper addition of shortcuts to a regular substrate can lead to the formation of a highly efficient structure for information propagation. Here we show that enhanced flow properties can also be observed in these complex 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, $P_{ij}\sim r_{ij}^{-α}$, where $r_{ij}$ 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 $g_{ij}\propto r_{ij}^{-δ}$, where $δ$ determines the extent of the geographical limitations (costs) on the long-range connections. Our results show that the best flow conditions are obtained for $δ=0$ with $α=0$, while for $δ\gg 1$ the overall conductance always increases with $α$. For $δ\approx 1$, $α=d$ 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.

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