Optimal spatial transportation networks where link costs are sublinear in link capacity

Consider designing a transportation network on n vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length l and capacity c scales as lc(beta) for fixed 0 < beta < 1. Under appropriate standardization, the cost of the minimum cost Gilbert network grows essentially as n alpha beta), where alpha(beta) = 1 - (beta/2) on 0 < beta = 1/2 and alpha(beta) = 1/2 + (beta/2) on 1/2 = beta < 1. This quantity is an upper bound in the worst case ( of vertex positions) and a lower bound under mild regularity assumptions. Essentially the same bounds hold if we constrain the network to be efficient in the sense that average route length is only 1 + o(1) times the average straight line length. The transition at beta = 1/2 corresponds to the dominant cost contribution changing from short links to long links. The upper bounds arise in the following type of hierarchical networks, which are therefore optimal in an order-of-magnitude sense. On the large scale, we use a sparse Poisson line process to provide long-range links. On the medium scale, we use hierarchical routing on the square lattice. On the small scale, we link vertices directly to medium-grid points. We discuss one of the many possible variant models, in which links also have a designed maximum speed s and the cost becomes lc(beta) s(gamma).