In recent years, as large datasets on urban areas around the world have become available, researchers like Bettencourt and Yang have begun analyzing scaling behaviors1that emerge in human systems—including cities. The field really ignited about a decade ago, she says, when researchers from SFI first showed that many properties of cities also changed in a predictable way over orders of magnitude in city size.

"There was this mysterious phenomenon that the properties of cities change in systematic ways with its size," says Yang. "That included things like fewer gas stations per capita, and a boost in socioeconomic activity, such as more research and development." Since then, researchers have found that many interesting socioeconomic properties increase disproportionally fast with population, said to be "superlinear." Some others grow disproportionally slowly and are said to be "sublinear." 

  • 1. Such scaling behavior has been found in systems ranging from hunter-gatherer societies to modern companies. The new framework offers a way to better understand and quantify properties with systematic trajectories—and even understand which ones contribute to the health of human institutions. It could, for example, give researchers a way to analyze how a phenomenon like economic growth changes with time and with population size (but does so along both dimensions in different ways). Bettencourt likens the new work to a Rosetta Stone that allows researchers to translate their findings between the two types of scaling.

Luís M. A. Bettencourt et al. The interpretation of urban scaling analysis in time, Journal of The Royal Society Interface (2020). 

DOI: 10.1098/rsif.2019.0846

Scaling is a general analytical framework used by many disciplines—from physics to biology and the social sciences—to characterize how population-averaged properties of a collective vary with its size. The observation of scale invariance over some range identifies general system types, be they ideal gases, ecosystems or cities. The use of scaling in the analysis of cities quantifies many of their arguably fundamental general characteristics, especially their capacity to create interrelated economies of scale in infrastructure and increasing returns to scale in socio-economic activities. However, the measurement of these effects, and the relationship of observable parameters to theory, hinge on how scaling analysis is used empirically. Here, we show how two equivalent approaches to urban scaling—cross-sectional and temporal—lead to the measurement of different mixtures of the same fundamental parameters describing pure scale and pure temporal phenomena. Specifically, temporal exponents are sensitive to the intensive growth of urban quantities and to circumstances when population growth vanishes, leading to instabilities and infinite divergences. These spurious effects are avoided in cross-sectional scaling, which is more common and closer to theory in terms of quantitative testable expectations for its parameters.