The interpretation of urban scaling analysis in time

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.