Structure, Scaling and Phase Transition in the Optimal Transport Network

Details Structure, Scaling and Phase Transition in the Optimal Transport Network Structure, Scaling and Phase Transition in the Optimal Transport Network

2006-07-31

arxiv.org/abs/cond-mat/0607819v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 3 figures

Scientific paper

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.

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