Structure, Scaling and Phase Transition in the Optimal Transport Network

TL;DR

This paper investigates the optimal design of electrical networks by minimizing dissipation under constraints, revealing phase transitions and scaling relations that inform network efficiency and topology.

Contribution

It establishes explicit scaling relations between currents and conductances and links to previous models, highlighting phase transitions in network topology.

Findings

Discontinuous change in network topology at phase transition

Scaling relations between currents and conductances

Optimal networks derive from a potential function

Abstract

We minimize the dissipation rate of an electrical network under a global constraint on the sum of powers of the conductances. We construct the explicit scaling relation between currents and conductances, and show equivalence to a a previous model [J. R. Banavar {\it et al} Phys. Rev. Lett. {\bf 84}, 004745 (2000)] optimizing a power-law cost function in an abstract network. We show the currents derive from a potential, and the scaling of the conductances depends only locally on the currents. A numerical study reveals that the transition in the topology of the optimal network corresponds to a discontinuity in the slope of the power dissipation.