A New Model for Urban Scaling

As the populations of cities grow, cities obey certain regularities. The economic output of a city grows in a certain way (faster than linearly, or superlinearly), for example, and others change in other ways, such as the volume of roads in a city which happen to grow in the opposite fashion (sublinearly). These empirical findings were published by Luís Bettencourt, Geoffrey West, and colleagues in 2007. Well, Bettencourt has just published a new mathematical model for why these properties of cities behave in the way they do.

This new article, "The Origins of Scaling in Cities," which appears in Science this week , has a simple mathematical model of a city: it's a combination of a social network of interactions and something actually embedded in physical space, which of course, is quite reasonable as that is what a city is—an agglomeration of people in a specific region.

From the press release:

"A city is first and foremost a social reactor," Bettencourt explains. "It works like a star, attracting people and accelerating social interaction and social outputs in a way that is analogous to how stars compress matter and burn brighter and faster the bigger they are."

This, too, is an analogy though, because the math of cities is very different from that of stars, he says.

Cities are also massive social networks, made not so much of people but more precisely of their contacts and interactions. These social interactions happen, in turn, inside other networks – social, spatial, and infrastructural – which together allow people, things, and information to meet across urban space.

While the actual paper contains a huge amount of math, Bettencourt's model is based on four simple assumptions (with my glosses in brackets):

(1) Mixing population. The city develops so that citizens can explore it fully given the resources at their disposal. [a city should be able to be traversed by its inhabitants]

(2) Incremental network growth. This assumption requires that infrastructure networks develop gradually to connect people as they join, leading to decentralized networks [cities grow a little bit at a time, and continue to "work"]

(3) Human effort is bounded ... The increasing mental and physical effort that growing cities can demand from their inhabitants has been a pervasive concern to social scientists. Thus, this assumption is necessary to lift an important objection to any conceptualization of cities as scale-invariant systems. [as a city grows, our ability to deal with it can't increase beyond some reasonable amount]

(4) Socioeconomic outputs are proportional to local social interactions ... From this perspective, cities are concentrations not just of people, but rather of social interactions.

If you quantify these assumptions in a certain way, it turns that you can not only explain the sublinear and superlinear behaviors of various urban properties, but you can also predict other features of cities, from land area to land rents. And it turns out that the model works really well! Here's a table showing these:

While this model does not explain regional aspects of cities or the disparities within an urban environment, as Bettencourt notes—it is ultimately about the city as a whole—it is a powerful means of understanding how cities, and the people within them, operate. Ultimately, cities are an instantiation of the optimal configuration for social interactions, and this is a great quantitative exploration of how they work.

Top image:Ron Henry/Flickr/CC