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To analyze the transport of information or material from a source to everynode of a network we use two quantities introduced in the study of rivernetworks: 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 $C_{0,max}\propto KM$. Fromnumerical simulations for spanning tree networks with scale-free topology andwith $10^2$ up to $10^7$ nodes, it is found, for large $K$, that the averagenumber 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 theones obtained from a mean-field approach. If the network is characterized by adegree distribution of connectivity $P(k)\propto k^{- \gamma}$, we also findthat the transport efficiency increases as long as $\gamma$ decreases and thatspanning tree networks with scale-free topology are more optimized to transferinformation or material than random networks.

Cost of material or information flow in complex transportation networks