On the Meaning of Urban Scaling

Abstract

Urban scaling laws describe how an urban quantity $Y$ varies with city population $P$, typically as $Y \sim P^β$. These relations are usually obtained from cross-sectional comparisons across cities at a given time (transversal scaling), but their link to the temporal evolution of individual cities (longitudinal scaling) remains unclear. Here we derive explicit expressions for the transversal exponent from the longitudinal dynamics of cities. We show that the measured exponent does not directly reflect individual city dynamics, but instead arises from a snapshot of a heterogeneous ensemble of cities with distinct growth trajectories. As a result, transversal scaling combines intrinsic dynamics with statistical effects due to the distribution of city sizes and correlations between population and city-specific parameters. Consequently, cross-sectional scaling laws cannot, in general, be used to infer the dynamics of individual cities. In particular, apparent sub- or superlinear scaling can emerge even when all cities follow linear longitudinal dynamics, as we demonstrate for the area-population relation. Strikingly, the behavior associated with the transversal exponent is in general not observed in the trajectory of any individual city, underscoring its collective, rather than dynamical, nature. More broadly, the transversal exponent has a clear dynamical meaning only under restrictive conditions-when cities behave as scaled versions of one another and path dependence is weak. Outside of these limits, it is not a law of urban growth, but a statistical artefact of heterogeneity.

Figures (4)

• Figure 1: Examples of longitudinal linear dependency between area ($A$) and population ($P$). a) For four cities of the WSFEvo dataset wsfevo studied in marquis2025universal and b) for nine cities in the angel2012atlas dataset. On each panel, the points represent the data (annual for the WSFEvo dataset, sporadic for the angel2012atlas dataset). For each city, the inverse density ($a$), the intercept ($b$) and the goodness-of-fit ($R^2$), estimated by ordinary least-square regression, are reported for each city. The line on each panel represent the best linear fit $A = a P + b$.
• Figure 2: Transversal scaling: area--population. a) Temporal evolution of the exponent $\beta_T$ for the 19 cities studied in marquis2025universal, estimated using OLS in log--log space. Error bars indicate 95% confidence intervals. b) Cross-sectional relation in 1985. The dashed line shows the best fit, with $\beta_T = 0.71$ ($R^2 = 0.78$). The population spans more than 2.5 orders of magnitude. Alternative estimators (Theil--Sen, weighted least squares) yield similar results (see Supplementary Material).
• Figure 3: Statistical contributions to the transversal exponent. On the data discussed in Fig. \ref{['fig:areapop']}, the local longitudinal exponent $\beta_i(t)$ is measured for each city. Its average $\langle \beta\rangle$ is shown in the orange curve. The two other contributions (red and green curve), introduced in Eq. \ref{['eq:betaT_cov_identity']}, are of opposite sign and the deviation between $\beta_T$ and $\langle\beta \rangle$ emerges from their non-null compensation. Inset: comparison between the measured exponent (done by directly fitting the data) and the predicted exponent given by Eq. \ref{['eq:betaT_cov_identity']}.
• Figure 4: Wages: transversal against longitudinal scaling. a) Evolution of wages as a function of city size. Each city is represented by contiguous colored points. There are 363 MSAs, with data for each year between 1969 and 2015. b) Measure of the transversal exponent (using OLS in log-log coordinates) over years. Error bars represent the 95% confidence interval. Data: from bettencourt2013origins. c) Decomposition of the contributions to the transversal exponent, discussed in Sec. \ref{['subsec:general']}. It is important to note the large deviation between the average longitudinal exponent and the transversal exponent. d) Comparison between predicted and measured transversal exponents. Red line: identity.