Clouds and building in the city of London. (Photo: zodebala/istock)

New research in the Journal of the Royal Society Interface reveals the geometry behind predictable scaling relationships that apply to cities worldwide. 

The paper, by Carlos Molinero and SFI External Professor Stefan Thurner of Complexity Science Hub Vienna, explains the fractal origins of two types of urban scaling laws, first documented by SFI researchers in 2007

The first law, “sublinear scaling,” is for systems that deliver resources. It means a city with a large population needs only ~80% as many roads, power lines, and gas stations per person as a city half its size. The second, “superlinear scaling,” applies to outputs of socioeconomic activity. It means a large city produces ~120% more wealth, patents, crime, pollution, and disease per person than a city half its size. 

“Remarkably, these two growth rates, 0.8 and 1.2., are showing up over and over again in literally dozens of city-related contexts and applications,” wrote Complexity Science Hub Vienna in a press release. “However, so far it is not really understood where these numbers come from.”
To investigate, Molinero and Thurner used open data on the height of buildings from more than 4,700 cities in Europe to map where people live. The scientists assigned a dot to every person living in a building. Together, these dots form sort of a "human cloud”- - a self-similar fractal within a city.

Using the human cloud, the researchers were then able to determine the fractal dimension of a city's population: They retrieved a number that describes the human cloud in every city. Similarly, they calculated the fractal dimension of cities' road networks. 

"Although these two numbers vary widely from city to city, we discovered that the ratio between the two is a constant," Thurner says. "It's this scaling exponent that determines how the properties of a city change with its size, and that is relevant because many cities around the world are growing rapidly.”

Read the paper, "How the geometry of cities determines urban scaling laws," in the Journal of the Royal Society Interface (March 17, 2021)

Read the Complexity Science Hub Vienna press release (March 17, 2021)