A long-time limit of world subway networks

Abstract: 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 which shares similar generic features} despite their geographic 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%. 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 barycenter 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.