Flow optimization process in a transportation network

Numerous networks, such as transportation, distribution and delivery networks optimize their designs in order to increase efficiency and lower costs, improving the stability of its intended functions, etc. Networks that distribute goods, such as electricity, water, gas, telephone and data (Internet), or services as mail, railways and roads are examples of transportation networks. The optimal design fixes network architecture, including clustering, degree distribution, hierarchy, community structures and other structural metrics. These networks are specifically designed for efficient transportation, minimizing transit times and costs. All sorts of transportation networks face the same problem: traffic congestion among their channels. In this work we considered a transportation network model in which we optimize/minimize a cost function for the flux/current at each channel/link of the network. We performed simulations and an analytical study of this problem, focusing on the fraction of used channels and the flow distribution through these channels. Our results show that, after the initial transient, the fraction of used channels stays constant and, remarkably, this result does not depend on the lattice structure (2D, 3D, or long-range connections). For the case of high flow, all channels in the network are used. On the other hand, in the small flow limit, we observe a novel behavior that the fraction of used channels depends on the square root of the flow.