Dynamic roughening of cities driven by multiplicative noise

TL;DR

The paper introduces a physics-inspired stochastic framework for urban growth that links vertical expansion to multiplicative noise and GDP-driven drift, including spatial coupling that leads to KPZ-type roughening. A zero-dimensional geometric Brownian motion extended to a spatial lattice is mapped, via the Hopf-Cole transformation, to the Kardar-Parisi-Zhang (KPZ) equation in the continuum limit. Empirically, GBM describes vertical growth in fast-growing cities (e.g., China), reveals a universal stationary distribution for normalized height in the mean-field regime, and shows a tight coupling between mean height and GDP; the intra-urban variance scales with the mean height, and cities approach stationarity under strong coupling. The work also demonstrates KPZ-like roughness scaling with an exponent near 0.4 in 2+1 dimensions, supporting universality of non-equilibrium roughening across cities and providing a parsimonious framework for cross-scale urban dynamics.

Abstract

The evolution of urban landscapes is rapidly altering the surface of our planet. Yet, our understanding of the urbanisation phenomenon remains far from complete. A fundamental challenge is to describe spatiotemporal changes in the built environment. A dynamic theory of urban evolution should account for both vertical and horizontal city expansion, analogous to the dynamical behaviour of surface growth in physical and biological systems. Here we show that building-height dynamics in cities around the world are well described by a zero-dimensional geometric Brownian motion (GBM), where multiplicative noise drives stochastic fluctuations around a deterministic drift associated with economic growth. To account for intra-city correlations, we extend the GBM with spatial coupling, revealing how local interactions effectively mitigate noise-driven fluctuations and shape urban morphology. The continuum limit of this spatial model can be recasted into the Kardar-Parisi-Zhang (KPZ) equation and we find that empirical estimates of the roughness exponent are in the range of the KPZ prediction for most cities. Together, these results show that multiplicative noise, moderated by local interactions, governs the evolution of urban roughness, anchoring spatiotemporal city dynamics in a well-established statistical physics framework.

Figures (17)

• Figure 1: GBM growth of world cities. (a) World map of analysed cities colored by the mean logarithmic building-height growth rate $\gamma$; inset highlights East Asia. Guangzhou, China, shown as an illustrative case (highlighted by the black box in the inset). (b–d) GBM–data comparison for Guangzhou: (b) per-pixel building heights (grey) with empirical and GBM mean height overlaid (Eq. \ref{['eq:GBM_interactions_mean']}); (c) variance of heights, empirical vs GBM (Eq. \ref{['eq:GBM_variance']}); (d) kernel-density estimates of the height distribution for five benchmark years. (e,f) GBM parameter estimates for all East Asian cities analyzed where $R^{2}$ refers to empirical mean (f) and variance (g) compared to GBM. (g) Temporal lienar relation between city mean height $\overline{h}$ and city GDP ($\alpha,\beta$ constant determined for each city; one line represent one city.)
• Figure 2: City‑level GBM behaviour for China.(a) Time series of the mean building height $\overline{h}(t)$ for each city (grey) and the national average (black), compared with the GBM fit (red). (b) Corresponding height variance $\mathrm{Var}\!\bigl(h(t)\bigr)$ and its GBM prediction. (c) Kernel‑density estimates of the height distribution for five benchmark years (1993–2020). (d) National‑average GDP (black) with GBM fit (red), GDP for each city (grey; city level GDP data are only available for the main Chinese cities NBS2025). (e) Variance of GDP and GBM prediction. (f) KDEs of the GDP distribution for the same benchmark years. Inset boxes list the fitted GBM parameters $(\mu,\sigma)$ and coefficients of determination $R^{2}$ for mean and variance. The analysis extends earlier intra‑urban study to the inter‑city scale.
• Figure 3: GBM with local interactions simulation Results of Eq. \ref{['eq:GBM_interactions']}, initialized in 1993 with parameters $(\mu,\sigma,k)$ optimized to minimize the overall RMSE. Guangzhou building height data (a) compared with simulation results (b) at different time steps. (c) Distribution of per-city RMSE values across all Chinese and neighboring cities. The map shows the city locations and their average heights from the data. Panels (a,b) Background maps from OpenStreetMap OpenStreetMap.
• Figure 4: Growth regimes. (a) Regime diagram showing the competition between multiplicative noise and mean-field coupling in the stochastic growth model; (b) temporal evolution of variance versus mean; (c) time evolution of the relative fluctuation ratio $r^*(t) = w(t)/(w(t)+1)$; (d) distribution of $r$ from Eq. \ref{['eq:ratio_var_mean']} ($r = lim_{t\rightarrow\infty}r^*(t)$ in panel (c)). (e) Empirical value of $r$ computed via Eq. \ref{['eq:ratio_var_mean']} for cities worldwide. (f) PDFs of building heights computed from 5km-resolution data for 2020; (g) for the same cities, PDFs of the normalized height $u$, together with the theoretical PDF from Eq. \ref{['eq:stationary_u']}; (h) Jensen–Shannon divergence (base 2) between the empirical PDFs of $u$ in the last and previous year along with the annual slope of the JSD relative to the fitted stationary inverse-gamma distribution (negative indicates convergence) for cities larger then 1250$km^2$. Lower-left indicates strongest convergence; Panels (b–d and f-h) show results for for East Asian cities, cities are represented by a gray line (or a dot in panel h), except for several example cities highlighted in color.
• Figure 5: Multi–scale roughness and KPZ-type scaling across the study cities. (a) Root-mean-square roughness $w(\ell)=\sqrt{\langle(\phi-\langle\phi\rangle_{\ell})^{2}\rangle_{\ell}}$ versus box size $\ell$ on log--log axes. Thin gray curves show all cities; colored lines highlight examples cities. The roughness exponent $\alpha$ is estimated from the pre-plateau regime. A dashed black reference line (labelled KPZ) indicates $w(\ell)\propto \ell^{0.4}$. (b) Distribution of $\alpha$ across cities; vertical markers denote the highlighted cases. Roughness exponent computed from the data 30-m resoluation data from chen2025characterizing for year 2020.