Optimal Supply Networks - Complex Networks CSYS/MATH 303, Spring, 2011

5 of 86 Single source optimal supply the potential at the nodes by solving the system of linear equations ik ! P lRkl"Uk #Ul$, then the currents through the links Ikl are determined. We use these currents to determine a first approximation of the optimal conductivities on the basis of the scaling relation. Then, the currents are recalculated with this set of conductivities, and the scaling relation is reused for the next approximation. These steps are repeated until the values have converged. We check by perturbing the solution that it actually is a minimum of the dissipation, which was always the case. For all !> 1, independently of the initial conditions, the same conductivity distribution is obtained, which conforms to the analytical result of [6]: there exists a unique minimum which is therefore global. Furthermore, the distribution of "kl is ‘‘smooth,’’ varying only on a ‘‘macroscopic scale,’’ as show in Fig. 2(a). No formation of any particular structure occurs. However, the conductivity distribution is not isotropic. We can interpret the conductivity distribution as a discrete approximation of a continuous, macroscopic conductivity tensor (see also [10]). The smooth aspect of the distribution is conserved while approaching ! ! 1 while the local anisotropy increases, while the values of all "kl remain finite, even if they get very small. For ! ! 1:5 and Ndia ! 15, the conductivity distribution spreads already over eight decades and becomes still broader as ! ! 1%, in which limit the number of iteration steps diverges as the minima becomes less and less steep. ! ! 1 presents a marginal case. The results of the simulation suggest that the minimum is highly degenerate; i.e., there are a large number of conductivity distributions yielding the same minimal dissipation. For !< 1, the output of the relaxation algorithm is qualitatively different [Fig. 2(b)]. A large number of conductivities converge to zero so that no loop remains. The highly redundant network is transformed to a spanning tree topology and the currents are canalized in a hierarchical manner. This, too, is predicted by the analytical results [6]. In contrast to !> 1, the conductivity distribution cannot be interpreted as a discrete approximation of a conductivity tensor: for Ndia ! 1, the structure becomes fractal. For different initial conditions, the relaxation algorithm yields trees with different topologies: each local minima in the high-dimensional and continuous space of conductivities f"klg corresponds to a different tree topology. To find the global minima with !< 1, we search consequently in the (exponentially large) space of tree topologies using a Monte Carlo algorithm. (We start with some initial tree and then switch links without creating loops and without disconnecting a part of the network.) Note that for a tree topology, the currents do not depend on the values "kl and, using the scaling relation, one may directly write down the dissipation rate for a given tree; the iterative relaxation is not necessary here. This regime has been widely explored in the context of river networks [4,5,13,15], mainly for a set of parameters that corresponds, in our case, to ! ! 0:5. An example of a result n minimal dissipation tree structure is given in Fig. 2(c). Note also, that the scaling relations can be seen as some kind of erosion model: the more currents flows through a link, the better the link conducts [4]. The qualitative transition is reflected also quantitatively in the value of the minimal dissipation [Fig. 3(a)]. The points for !> 1 were obtained with the relaxation algorithm, the points !< 1 by optimizing the tree topologies with a Monte Carlo algorithm. For ! ! 1, Jmin=Jhomo ! 1 by definition; for ! ! 0, Jmin=Jhomo ! 0, because the vanishing "kl allow the remaining "kl ! 1. Figure 3(b) shows the behavior of minimal dissipation rate close to ! ! 1. For ! smaller than 1, the relaxation method only furnishes a local minimum, the Monte Carlo algorithm searching for the optimal tree topologies gives lower dissipation values. The different values corresponding to different realization indicate that the employed Monte Carlo method does not find the exact global minima. For !> 1, the optimal tree obtained by the Monte Carlo algorithm is not the optimal solution since the absolute and only minima has loops. The dissipation rate which results from the relaxation algorithm is then, of course, lower than the dissipation of any tree. While the curve is continuous, the crossover at ! ! 1 shows a clear change in the slope of Jmin"!$. One could interpret this behavior as a second order phase transition. (The change in