Flow optimization process in a transportation network

TL;DR

This paper studies flow optimization in transportation networks, revealing that the fraction of used channels stabilizes regardless of network structure and exhibits a square root relation to flow in low-flow conditions.

Contribution

It provides an analytical and simulation-based analysis of flow distribution and channel utilization in transportation networks, highlighting universal behaviors and novel flow-dependent phenomena.

Findings

Fraction of used channels stabilizes after initial transient.

Used channels depend on the square root of flow in low-flow limit.

All channels are used at high flow levels.

Abstract

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.