Bohn Steffen, Magnasco Marcelo O
Center for Studies in Physics and Biology, Rockefeller University, Box 212, 1230 York Avenue, New York, NY, USA.
Phys Rev Lett. 2007 Feb 23;98(8):088702. doi: 10.1103/PhysRevLett.98.088702. Epub 2007 Feb 21.
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)10.1103/PhysRevLett.84.4745]. 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.
最优网络的结构和属性取决于被最小化的成本函数以及最小化所受的约束条件。我们在此展示两种不同的公式表述，它们会得出相同的结果：在全局约束下最小化电网络的耗散率等同于最小化由巴纳瓦尔等人引入的幂律成本函数[《物理评论快报》84, 4745 (2000)10.1103/PhysRevLett.84.4745]。推导出了电流与相应电导之间的显式标度关系，证明了后者的势流性质。改变一个唯一的参数，优化网络的拓扑结构会显示出从树形拓扑到带有回路的非常冗余结构的转变；该转变对应于功率耗散斜率的不连续性。
