Cost of material or information flow in complex transportation networks

TL;DR

This paper investigates the cost and efficiency of transporting information or material in complex networks, revealing that scale-free topologies optimize transfer and that costs grow proportionally to network size and logarithm.

Contribution

It introduces a theoretical framework for analyzing transport costs in complex networks and demonstrates that scale-free networks are more efficient than random networks.

Findings

Global transport cost scales as K ln(K) for large networks.

Scale-free networks with lower gamma are more efficient.

Scale-free topologies outperform random networks in transfer efficiency.

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.