The interpretation of urban scaling analysis in time

Abstract

<jats:p>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.</jats:p>

Urban Scaling as Validation for Predictions of Imperviousness From Population

“…This approach identifies the empirical relationship between n and y that is based only on cross city variation, while still taking advantage of the temporal data that we have. It is consistent with the strategy described by Bettencourt et al (2020). Within city variation, or changes in the n , y , difference from expectation across time informs only our confidence intervals around estimates of α , β , or γ j .…”

“…Most of the urban scaling literature uses cross‐sectional data, which are data that only have spatial variation to explore urban scaling effects. More recently, explorations of urban scaling theory have begun to use panel data sets which include both spatial and temporal variation (Bettencourt et al, 2020; Depersin & Barthelemy, 2018). In this field, there has been little engagement thus far with well‐established techniques developed by economists and statisticians for the assessment of panel data.…”

Association between population distribution and urban GDP scaling

“…Fostered by a recent deluge of highly-detailed city data, researchers from diverse disciplines such as geography, economics, or physics have devoted ongoing efforts in identifying and understanding fundamental principles and regularities underlying urban systems [ 1 , 6 , 12 – 14 ]. While most of these works are either concerned with Zipf’s law [ 15 – 25 ] or urban scaling [ 26 – 34 ], very few have tackled the relationship between both. Zipf’s law and urban scaling have mostly been studied independently because it is commonly assumed that both laws are independent descriptions of urban systems across countries.…”

Hamiltonian modelling of macro-economic urban dynamics

“…The quasistationarity of the coupling parameters just mentioned, suggests the possibility to describe the evolution of urban macro-economic indicators as the solution of a stochastic differential equation of the Langevin type. Though in the literature cities are often described as out-of-equilibrium systems [5,20], we observe that assuming a quasi-equilibrium dynamical evolution of rescaled indicators allows to forecast with high precision their future value in individual cities. To this end, we assume that the vector of indicators obeys the solution of a discretized Langevin equation whose stationary state is given by the previously inferred Hamiltonian.…”