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, $β$ from $\langle N_\ell\rangle\sim \ell^β$ and $γ$ from $\mathrm{Var}(N_\ell)\sim \ell^γ$, where in the small-$\ell$ limit $β$ equals the planar fractal dimension of populated space. Across cities within a given year, $γ$ depends linearly on $β$. 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 $γ\simeq 2+β$. A mean-field (independent-cell) argument predicts a quadratic mean--variance mapping and cannot reproduce the observed $β$--$γ$ dependence, implying strong spatial correlations. We derive a correlation-aware variance decomposition in which $γ$ is controlled by a correlation dimension $D_c$; in the correlation-dominated regime $γ=2+D_c$. If large maturing cities, as are the ones selected in our dataset, evolve to effective monofractal ($D_c\simeq β$) cities, the asymptotic prediction becomes $γ\simeq 2+β$, 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.

Figures (8)

• Figure 1: Scaling analysis pipeline for grid-level population data. (A) Coarse-graining the population data by doubling the box size $\ell$ and aggregating inhabitants in each box. Colors indicate the number of residents within boxes of area $\ell^2$. (B) Top: example coarse-graining steps for Amsterdam, with administrative boundaries shown in black. Bottom: empirical distributions of $N_\ell$ for four values of $\ell$; grey points show city-level estimates, and black markers summarize the Netherlands-wide aggregation. (C) Mean and variance of $N_\ell$ for the city of Amsterdam as a function of $\ell$ in log--log coordinates; slopes define $\beta$ and $\gamma$ (Eq. \ref{['eq:beta_gamma_def']}). Data shown here are for the year 2023 cbs2024dataset. © Centraal Bureau voor de Statistiek, © ESRI Nederland.
• Figure 2: Scaling exponents for regions in the Netherlands. Exponents for the mean ($\beta$) and variance ($\gamma$) of inhabitants per box across Dutch urban regions (2023) show a strong linear relationship (Eq. \ref{['eq:hyperscaling_line']}). Points are colored by CBS urbanization level. Insets show two representative regions spanning the spectrum (low-density/low-$\beta$ vs. high-density/high-$\beta$). Error bars indicate uncertainties from the log--log fits used to estimate $\beta$ and $\gamma$ (see Supplementary Materials).
• Figure 3: Global hyperscaling across five decades (GHS-POP). Exponent pairs $(\beta,\gamma)$ for major world cities computed from GHS-POP at 5-year intervals from 1975--2020, pooled across cities (see Supplementary Materials for continent-level stratification). The selection of cities is kept constant for each year. As cities progress in their evolution, the scaling relationship between $\gamma$ and $\beta$ shows a drift toward $c_1\to2$ and $c_2\to1$, i.e. $\gamma\simeq2+\beta$.
• Figure S1: Coarse graining analysis for two cities from the GHS-POP dataset (Left) New-York City (USA) and (Right) Delhi (India).
• Figure S2: (Left) Hyperscaling parameters $c_1$ and $c_2$ for the equation $\gamma = c_1 + c_2 \beta$ for the subset of Dutch cities chosen for this study in different years. (Middle & Right) Hyperscale parameters $c_1$ and $c_2$ for cities grouped by their continents. The band indicates standard deviation from the linear regression of the inferred parameters.