Cost of material or information flow in complex transportation networks

Abstract

To analyze the transport of information or material from a source to every node of a network we use two quantities introduced in the study of river networks: the cost and the flow. For a network with $K$ nodes and $M$ levels, we show that an upper bound to the global cost is $C_{0,max}\propto KM$. From numerical simulations for spanning tree networks with scale-free topology and with $10^2$ up to $10^7$ nodes, it is found, for large $K$, that the average number of levels and the global cost are given by $M\propto \ln (K)$ and $C_0\propto K\ln (K)$, respectively. These results agree very well with the ones obtained from a mean-field approach. If the network is characterized by a degree distribution of connectivity $P(k)\propto k^{-\gamma }$, we also find that the transport efficiency increases as long as $\gamma$ decreases and that spanning tree networks with scale-free topology are more optimized to transfer information or material than random networks.

4 pages, 2 figures

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