The structure of branched transportation networks

The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure μ + onto a measure μ - with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s α with 0 < α < 1. For the Monge-Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow s1+s2 is preferable to two individual roads s 1 and s 2 because (s1 + s2)α < s1α+s2α. Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When α > 1-1/N such structures can irrigate a whole bounded open set of ℝN. The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia. © 2007 Springer-Verlag.