Hyperscaling of spatial fluctuations constrains the development of urban populations

TL;DR

This study investigates how spatial fluctuations and fractal organization in urban populations scale with city size, revealing a robust hyperscaling relation influenced by spatial correlations and urban development over time.

Contribution

It introduces a new hyperscaling relation between mean and variance of population distribution, accounting for spatial correlations and temporal evolution in urban systems.

Findings

The hyperscaling relation $ ext{Var}(N_ ext{ell}) o ext{mean}(N_ ext{ell})$ is robust but varies across regions and over time.

A correlation dimension $D_c$ controls the variance scaling, linking urban form to fluctuations.

Large cities tend to evolve toward monofractal structures, consistent with the $ ext{Var} o ext{mean}$ relation.

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