Universal roughness and the dynamics of urban expansion

Abstract

Urban sprawl reshapes cities, yet its quantitative laws remain elusive. Analyzing built-up expansion in 19 cities (1985-2015) with tools from surface growth physics in radial geometry, we reveal anisotropic, branch-like growth and a piecewise linear scaling between area and population. We uncover a robust local roughness exponent $α_{\text{loc}}\approx 0.54$, coexisting with variable $β$ and $z$. This unusual coexistence of universal and variable exponents offers a rare empirical testbed for nonequilibrium growth and an empirical basis for modeling urban sprawl.

Figures (4)

• Figure 1: Different growth patterns of the largest connected component (LCC) area vs. population (1985–2015): (A) linear scaling at constant density (Beijing, $\sim$3,685 inh./km$^2$); (B) piecewise linear growth with increasing density (Guatemala City, from 4,437 to 15,314 inh./km$^2$); (C) saturation plateau (Las Vegas, up to 16,782 inh./km$^2$). Inset: Historical trends (1800–2000). Beijing shows a density drop from 48,393 to 2,882 inh./km$^2$ around 7M population, while Guatemala City maintains near-constant density (7,576 inh./km$^2$).
• Figure 2: Relative aggregated area as a function of demographic pressure, exhibiting a clear increasing trend. The dashed line represents a power-law fit with an exponent of approximately $1.22$ (with $R^2=0.71$), provided as a visual guide. While the population growth rate changes from $\approx 1\%$ for London to $6\%$ for Ningbo, the relative size of aggregates increases by more than one and a half order or magnitude (the error bars represent the standard error). Inset: average ratio of coalesced built sites ($C_o$) relative to newly built sites ($C_n$). An increasing trend quantified by a Spearman's correlation of $0.51$ is observed. The dark dashed line represents a power-law of exponent $0.33$. Color code : geographic region.
• Figure 3: (A) Interface width $w$, computed for angular sectors $\Delta \theta$, plotted against the average arc length $\overline{\ell}=R\Delta\theta$ (where $R=\langle r\rangle_i$ for each sector $i$) for the city of Ningbo, China, for each year between 1985 (purple) and 2015 (red). (B) According to the scaling form in Eq. \ref{['eq:scaling']}, these curves should collapse onto a single master curve when plotting $wP^{-\beta}$ against $R\Delta\theta P^{-1/z}$. This collapse allows for the determination of the exponents $\beta$ and $z$.
• Figure 4: Exponent $\beta$ versus $1/z$ for cities in our dataset. We can distinguish 3 broad groups according to the value of $\beta$ (shown in different colors): purple for $\beta$ small, green for $\beta\approx 0.37$, and red for $\beta>0.6$. We also indicate usual universality classes values for $\beta$ and $z$: Edwards-Wilkinson (EW) with $\beta=1/4$, $z=2$ (and $\alpha=1/2$), Kardar-Parisi-Zhang (KPZ) with $\beta=1/3$, $z=3/2$ (and $\alpha=1/2$), and Mullins-Herring (MH) with $\beta=3/8$, $z=4$ (and $\alpha=3/2$), along with two quenched universality classes at the depinning transition, qKPZ ($\beta=0.63$, $z=1$) and qMH ($\beta=0.84$, $z=1.78$).