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

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(i<j)C(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 |I(ij)|(gamma) with 0 < gamma < 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.