Image caption The study found that networks converged to a dense core of central stations from which branches radiate

A study of the world's largest subway networks has revealed that they are remarkably mathematically similar.

The layouts seem to converge over time to a similar structure regardless of where or over how long they were built.

The study, in the Journal of the Royal Society Interface , analysed 14 subway networks around the world.

It found common distributions of stations within the networks, as well as common proportions of the numbers of lines, stations, and total distances.

In some senses, it is unsurprising that the study found that networks tended over time to comprise a dense core of central stations with a number of lines radiating outward from it.

By choosing the world's largest networks, from Beijing to Barcelona, the results were bound to represent networks that serve city centres with a dense collection of stations and bring commuters inward from more distant stations.

But the analysis shows a number of less obvious similarities across all 14 networks.

It found the total number of stations was proportional to the square of the number of lines - that is, a four-fold increase in station number would result in a doubling of the number of lines.

The dense core of central stations all had the same average number of neighbours in the network, and in all cases, about half the total number of stations were found outside the core.

In addition, the length of any one branch from the core's centre was about the same as twice the diameter of the core.

The number of stations at a given distance from the centre was proportional to the square of that distance, but only up to the edge of the core; at more distant reaches of the network, the number of stations contained was directly proportional to distance.

Image caption Slime moulds grow to seek "optimum" networks that parallel subway organisation

The authors analysed how the networks grew and added lines and stations, finding that they all converged over time to these similar structures.

They authors point out that the similarities exist regardless of where the networks were, when they were begun, or how quickly they reached their current layout.

"Although these (networks) might appear to be planned in some centralised manner, it is our contention here that subway systems like many other features of city systems evolve and self-organise themselves as the product of a stream of rational but usually uncoordinated decisions taking place through time," they wrote.

The authors say that the systems do not appear to be "fractal". Fractal systems follow mathematical patterns that seem equivalent in a number of physical and social systems ranging from the movements of planets to the movements of depressed people, but they may or may not reflect a deeper, more universal organisational principle .

Nevertheless, the team wrote that some underlying rule is likely to be driving the way subway systems end up worldwide.

"The existence of unique long-time limit topological and spatial features is a universal signature that fundamental mechanisms, independent of historical and geographical differences, contribute to the evolution of these transportation networks," they wrote.