CAMS (CNRS/EHESS) 190, avenue de France, 75013 Paris, France.
J R Soc Interface. 2012 Oct 7;9(75):2540-50. doi: 10.1098/rsif.2012.0259. Epub 2012 May 16.
We study the temporal evolution of the structure of the world's largest subway networks in an exploratory manner. We show that, remarkably, all these networks converge to a shape that shares similar generic features despite their geographical and economic differences. This limiting shape is made of a core with branches radiating from it. For most of these networks, the average degree of a node (station) within the core has a value of order 2.5 and the proportion of k = 2 nodes in the core is larger than 60 per cent. The number of branches scales roughly as the square root of the number of stations, the current proportion of branches represents about half of the total number of stations, and the average diameter of branches is about twice the average radial extension of the core. Spatial measures such as the number of stations at a given distance to the barycentre display a first regime which grows as r(2) followed by another regime with different exponents, and eventually saturates. These results--difficult to interpret in the framework of fractal geometry--confirm and yield a natural explanation in the geometric picture of this core and their branches: the first regime corresponds to a uniform core, while the second regime is controlled by the interstation spacing on branches. The apparent convergence towards a unique network shape in the temporal limit suggests the existence of dominant, universal mechanisms governing the evolution of these structures.
我们以探索的方式研究了世界上最大的地铁网络的结构随时间的演变。我们发现，令人惊讶的是，尽管这些网络在地理和经济上存在差异，但它们都收敛到一种具有相似通用特征的形状。这种限制形状由一个核心和从核心辐射出的分支组成。对于大多数这些网络，核心内一个节点（车站）的平均度数为 2.5 左右，核心内 k=2 节点的比例大于 60%。分支的数量大致与车站数量的平方根成正比，当前分支的比例约占总车站数量的一半，分支的平均直径约为核心平均径向延伸的两倍。在给定距离到质心的车站数量等空间度量上，呈现出一种首先以 r(2)增长的模式，然后是具有不同指数的另一种模式，最终趋于饱和。这些结果——在分形几何的框架下很难解释——在这个核心及其分支的几何图像中得到了证实和自然解释：第一个模式对应于一个均匀的核心，而第二个模式则由分支上的站间间隔控制。在时间限制下，这些结构似乎朝着独特的网络形状收敛，这表明存在主导的、普遍的机制来控制这些结构的演化。
