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

The structure and properties of optimal networks depend on the cost functional being minimized and on constraints to which the minimization is subject. We show here two different formulations that lead to identical results: minimizing the dissipation rate of an electrical network under a global constraint is equivalent to the minimization of a power-law cost function introduced by Banavar et al. [Phys. Rev. Lett. 84, 4745 (2000)]. An explicit scaling relation between the currents and the corresponding conductances is derived, proving the potential flow nature of the latter. Varying a unique parameter, the topology of the optimized networks shows a transition from a tree topology to a very redundant structure with loops; the transition corresponds to a discontinuity in the slope of the power dissipation.

Figure 1

Sketch of a loop $\alpha$ indicating the direction of the currents. Every perturbation of the $I_{kl}$ satisfying the constraints can be written as a weighted sum of such loops.

Figure 2

Examples of the optimized conductivity distributions obtained by the relaxation method for (a) $\gamma =2.0$ and (b) $\gamma =0.5$. For $\gamma <1$, the relaxation leads in general only to a local minimum. The global minimum can be approached by searching in the space of tree topologies. The result for $\gamma =0.5$ is shown in (c). The arrows indicate the localized inlet.

Figure 3

(a) The normalized minimum dissipation rate $J_{\min }/J_{\mathrm{homo}}$ as a function of $\gamma$ for a network with $N_{\mathrm{dia}}=15$ (462 links) and a network with $N_{\mathrm{dia}}=31$ (2070 links, in red or in gray). Note the discontinuity of the slopes at $\gamma =1$. (b) A detailed view of the crossover at $\gamma =1$ for $N_{\mathrm{dia}}=15$. Cross symbols show data points obtained by optimizing a tree topology, circles show the output of the relaxation algorithm. The continuous lines indicate the actual minimum.