The growth equation of cities

TL;DR

This paper introduces a new stochastic model for city population growth based on empirical data, revealing that Zipf's law does not universally apply and highlighting the significance of rare migratory shocks in urban evolution.

Contribution

The paper develops a novel stochastic equation for city growth that accounts for rare migratory shocks, challenging the universality of Zipf's law in urban systems.

Findings

Zipf's law does not hold universally due to finite-time effects.

Rare migratory shocks significantly influence city growth dynamics.

City hierarchies exhibit multiple temporal variations.

Abstract

The science of cities seeks to understand and explain regularities observed in the world's major urban systems. Modelling the population evolution of cities is at the core of this science and of all urban studies. Quantitatively, the most fundamental problem is to understand the hierarchical organization of cities and the statistical occurrence of megacities, first thought to be described by a universal law due to Zipf, but whose validity has been challenged by recent empirical studies. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations, and despite many attempts these fundamental questions have not been satisfactorily answered yet. Here we fill this gap by introducing a new kind of stochastic equation for modelling population growth in cities, which we construct from an empirical analysis of recent datasets (for Canada, France, UK and USA) that reveals how rare but large interurban migratory shocks dominate city growth. This equation predicts a complex shape for the city distribution and shows that Zipf's law does not hold in general due to finite-time effects, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations. Our result underlines the importance of rare events in the evolution of complex systems and at a more practical level in urban planning.