Universal Scaling of Optimal Current Distribution in Transportation Networks

TL;DR

This paper demonstrates that in d-dimensional transportation networks, the optimal current distribution follows a universal power-law decay, independent of specific convex cost functions, revealing scale-invariance properties.

Contribution

It proves the scale-invariance of optimal current distributions in d-dimensional networks and derives the exact distribution for two-dimensional cases.

Findings

Current distribution decays as a power law with exponent (2d-1)/(d-1).

Scaling properties are robust under random injections and cost function variations.

Exact solutions are obtained for two-dimensional networks.

Abstract

Transportation networks are inevitably selected with reference to their global cost which depends on the strengths and the distribution of the embedded currents. We prove that optimal current distributions for a uniformly injected d-dimensional network exhibit robust scale-invariance properties, independently of the particular cost function considered, as long as it is convex. We find that, in the limit of large currents, the distribution decays as a power law with an exponent equal to (2d-1)/(d-1). The current distribution can be exactly calculated in d=2 for all values of the current. Numerical simulations further suggest that the scaling properties remain unchanged for both random injections and by randomizing the convex cost functions.