Optimal transportation network with concave cost functions: loop analysis and algorithms.

TLDR: A loop analysis of transportation cost is performed, and an alternative mathematical proof of the optimality of tree-formed networks is given, which leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions.

Abstract: Transportation networks play a vital role in modern societies. Structural optimization of a transportation system under a given set of constraints is an issue of great practical importance. For a general transportation system whose total cost C is determined by C=Sigma C-i < j(ij)(I-ij), with C-ij (I-ij) being the cost of the flow I-ij between node i and node j, Banavar and co-workers [Phys. Rev. Lett. 84, 4745 (2000)] proved that the optimal network topology is a tree if C-ij proportional to parallel to I-ij parallel to(gamma) with 0 < 1. The same conclusion also holds in the more general case where all the flow costs are strictly concave functions of the flow I-ij. To further understand the qualitative difference between systems with concave and convex cost functions, a loop analysis of transportation cost is performed in the present paper, and an alternative mathematical proof of the optimality of tree-formed networks is given. The simple intuitive picture of this proof then leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions.