Modeling the spatial growth of cities

TL;DR

The paper surveys mathematical approaches to spatial urban growth, framing urban sprawl as a complex, multi-mechanism process distinct from population growth alone. It assembles insights from geography, economics, and statistical physics, highlighting empirical regularities (density decay, fractality, vertical growth) and diverse modeling frameworks: cellular automata and agent-based models, microeconomic land-use theories (AMM and extensions), and a suite of statistical-physics-inspired growth models (DLA, Eden, percolation, MRFs, and dispersal/coalescence constructs). A central theme is coevolution and the role of transport infrastructure as an external field that reshapes urban form, with key results showing how coalescence, anisotropy, and network feedbacks drive multi-centered, uneven, and fractal-like urban morphologies. The review identifies open problems and directions for integrating first-principles dynamics with data-rich empiricism, aiming to develop predictive, policy-relevant tools for sustainable urban planning. Overall, the work emphasizes interdisciplinary synthesis to understand and forecast the spatio-temporal evolution of cities amid increasing data availability and shifting infrastructural landscapes.

Abstract

The growth of cities has traditionally been studied from a population perspective, while urban sprawl - its spatial growth - has often been approached qualitatively. However, characterizing and modeling this spatial expansion is crucial, particularly given its parallels with surface growth extensively studied in physics. Despite these similarities, approaches to urban sprawl modeling are fragmented and scattered across various disciplines and contexts. In this review, we provide a comprehensive overview of the mathematical modeling of this complex phenomenon. We discuss the key challenges hindering progress and examine models inspired by statistical physics, economics and geography, and theoretical ecology. Finally, we highlight critical directions for future research in this interdisciplinary field.

Figures (65)

• Figure 1: Example of urban sprawl: Growth of Changzhou (China) between 1985 and 2015 (data from wsfevo). We notice several seeds present before 1985 (in light blue), connected later. The main component exhibits a branch diagonally from northwest-west. Density decreases anisotropically with distance to the giant component.
• Figure 2: Illustration of urban sprawl in the city of London from 1800 to 2013. Data for the period 1800-1978 are from angel2012atlas and for 2000 and 2013 from angel2016atlas (see also AtlasUrbanExpansionHistorical for a video documenting the historical evolution of London and many other cities worldwide).
• Figure 3: Illustration of the quantity $r(\theta)$ for the frontier of the giant component of a city lemoy2021marquis2025universalroughnessdynamicsurban.
• Figure 4: Illustration of the CCA algorithm. To identify urban clusters, the City Clustering Algorithm (CCA) considers as connected all adjacent grid cells with nonzero population (in blue). The process begins by selecting an arbitrary populated cell, in the red seed in the top-right corner figure. (the final outcome is independent of this initial choice). A cluster is then grown iteratively by including all nearest neighbors of the current boundary that also have strictly positive population. This continues until no further neighboring populated cells remain. The procedure is repeated for each remaining unvisited populated cell until all such cells are assigned to a cluster. Adapted from rozenfeld2008laws.
• Figure 5: Connected components (CC) of built-up area in the Tokyo urban area, 1985. Connected components of multiple scales are visible : the giant cluster in light brown is several orders of magnitude larger than the any other clusters of macroscopic sizes, which are themselves much larger than the multitude of microscopic clusters. Each CC is assigned a random color to facilitate their visualization. Data from wsfevo.