Hyperscaling of spatial fluctuations constrains the development of urban populations

Abstract

Urban populations exhibit fractal organization and systematic scaling regularities, yet the scaling exponents reported across cities vary substantially, challenging existing theory. Using 100~m gridded population maps for 477 urban areas spanning the Netherlands (2000--2023) and major world cities (1975--2020), we recursively coarse-grain each city and quantify how the mean and variance of inhabitants in square grid cells of side length $\ell$ scale with $\ell$. This yields two exponents, ${\hat{I}}^2$ from $⟨N_{\ell }⟩\sim {\ell }^2$ and ${\hat{I}}^3$ from $\mathrm{Var}(N_{\ell })\sim {\ell }^3$, where in the small-$\ell$ limit ${\hat{I}}^2$ equals the planar fractal dimension of populated space. Across cities within a given year, ${\hat{I}}^3$ depends linearly on ${\hat{I}}^2$. Compiling $>$10,000 exponent estimates over five decades shows that this hyperscaling relation is robust yet non-universal: its slope and intercept vary across continents and drift systematically in time, trending toward the limiting form ${\hat{I}}^3\simeq 2+{\hat{I}}^2$. A mean-field (independent-cell) argument predicts a quadratic mean--variance mapping and cannot reproduce the observed ${\hat{I}}^2$--${\hat{I}}^3$ dependence, implying strong spatial correlations. We derive a correlation-aware variance decomposition in which ${\hat{I}}^3$ is controlled by a correlation dimension $D_c$; in the correlation-dominated regime ${\hat{I}}^3=2+D_c$. If large maturing cities, as are the ones selected in our dataset, evolve to effective monofractal ($D_c\simeq {\hat{I}}^2$) cities, the asymptotic prediction becomes ${\hat{I}}^3\simeq 2+{\hat{I}}^2$, consistent with the observed temporal drift. This interdependence links urban form and fluctuations, constrains mechanistic growth models, and implies scaling predictions for spatial indicators built from local means and variances.

39 pages, 8 figures, 3 tables