After decades of urban evolution, the world's major subway systems appear to be converging on an ideal form.

On the surface, these core-and-branch systems – evident in New York City, Tokyo, London or most any large metropolitan subway – may seem intuitively optimal. But in the absence of top-down central planning, their movement over decades toward a common mathematical space may hint at universal principles of human self-organization.

Understand those principles, and one might "make urbanism a quantitative science, and understand with data and numbers the construction of a city," said statistical physicist Marc Barthelemy of France's National Center for Scientific Research.

In a May 15 Journal of the Royal Society Interface paper, Barthelemy and NCSR complex systems analyst Camille Roth focused a network analysis lens on the aforementioned cities' subways, along with Barcelona, Beijing, Berlin, Chicago, Madrid, Mexico, Moscow, Osaka, Paris, Seoul and Tokyo.

With equations used to study two-dimensional spatial networks, the class of network to which subways belong, the researchers turned stations and lines to a mathematics of nodes and branches. They repeated their analyses with data from each decade of a subway system's history, and looked for underlying trends.

Patterns emerged: The core-and-branch topology, of course, and patterns more fine-grained. Roughly half the stations in any subway will be found on its outer branches rather than the core. The distance from a city's center to its farthest terminus station is twice the diameter of the subway system's core. This happens again and again.

"Many other shapes could be expected, such as a regular lattice," said Barthelemy. "What we find surprising is that all these different cities, on different continents, with different histories and geographical constraints, lead finally to the same structure."

Subway systems seem to gravitate towards these ratios organically, through a combination of planning, expedience, circumstance and socioeconomic fluctuation, say the researchers.

This is a crucial point: If the subways followed a predetermined path, their evolution would only reflect a set plan. Instead, the convergence "is a sign that there are some basic, profound mechanisms that drive the development of urban systems," said Barthelemy.

City subway systems converge on a ratio for the number of stations on branch lines to the number in city cores. Image: Roth et al./JRSI

According to Andrew Adamatzky, a University of West England computer scientist who uses creatures called slime molds to study optimal transportation networks, it's possible that engineers were influenced by early subway networks in London, Berlin and Paris.

But Adamatzky still called the results "very promising," saying that "when more data will be collected, then maybe they can suggest some useful theory of subway development."

Barthelemy's group says the trends they observed could next be cross-referenced with social shifts in cities. Their ultimate goal is a model of subway evolution based on real-world observations. With such a model, they could look for ways to tweak future transportation systems in optimal ways.

To be sure, this wouldn't be the first time that urbanists have tried to impose rationality and order on cities. The fruits of some such efforts – such as the vast, soulless housing projects of beloved of rectangle-obsessed modernists – proved bitter indeed.

But Barthelemy said his group's approach is different, seeking to improve on how people naturally organize themselves rather imposing abstract, arbitrary rules from above.

"We don't have big ideas," Barthelemy said. "We're just in the process of trying to understand centuries of development."

Citation: "A long-time limit for world subway networks." By Camille Roth, Soong Moon Kang, Michael Batty and Marc Barthelemy. Proceedings of the Royal Society Interface, 15 May 2012.