Long-time limit of world subway networksCamille Roth, Soong Moon Kang, Michael Batty, Marc Barthelemy

We study the temporal evolution of the structure of the world’s largest subway networks.

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 the core has a value of order 2.5, slowly increases with time and displays small fluctuations. The current proportion of branches represents about 40% 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 r to the barycenter display a first regime growing as r^2 followed by another regime with different exponents. These results — which are difficult to interpret in the framework of fractal geometry — confirm and find a natural explanation in the core and branches picture: the first regime corresponds to a uniform core, while the second regime is controlled by the interstation spacing on branches. The existence of a unique network shape in the temporal limit suggests the existence of dominant, universal mechanisms governing the evolution of these structures.

via [1105.5294] Long-time limit of world subway networks. [-]

Another excerpt that caught my attention:

Indeed, for some of the networks — such as Moscow, Seoul, and Shanghai — we observe larger differences with respect to the average, limiting network.

In the case of Moscow, its core appears over-developed compared to its branches. This network has resulted basically from a well-dened design and it is expected that it does not follow the same rules that govern networks evolving over a longer period which often appear to evolve in as lightly more self-organized manner. In the case of Seoul and Shanghai, it seems that their relatively young age could explain why they have not yet reached the longtime limit. We can note here that the least expensive way for these almost mature networks to reach the well-balanced long time limit is by constructing the minimum number of stations and lines and this thensuggests that Seoul and Shanghai need to increase their core density hence the value of (t) (by adding inter-branches links for example) and Moscow needs to increase the number of its branches. It will be interesting to observe their future evolution.