Cost of material or information flow in complex transportation networks
Centro de Formação de Professores, Universidade Federal do Recôncavo da Bahia - 45.300-000, Amargosa, BA, Brazil
2 Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais - C. P. 702, 30161-970, Belo Horizonte, MG, Brazil
Accepted: 30 April 2010
To analyze the transport of information or material from a source to every node of a network, in a steady-state situation, we use two quantities introduced in the study of river networks: the cost and the flow. We study a network with K+1 nodes (the source plus K nodes) and M levels. The level of a node is defined as the number of links between the source and the node. We show that an upper bound to the global cost is C0,max KM. From numerical simulations for spanning-tree networks with scale-free topology and with 102 up to 107 nodes, it is found, for large K, that the average number of levels, the average level of the nodes, M , and the global cost are given by M ln(K), M ln(K) and C0 K ln(K), respectively. These asymptotic 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) k-γ, we also find that the transport efficiency increases as long as γ decreases and that spanning-tree networks with scale-free topology are more optimized to transfer information or material than random networks.
PACS: 05.10.-a – Computational methods in statistical physics and nonlinear dynamics / 05.45.-a – Nonlinear dynamics and chaos / 87.18.Sn – Neural networks and synaptic communication
© EPLA, 2010