Scientific result | Theoretical physics | Environment

# Modeling the city: an equation that settles a century-old controversy

*Published on 18 November 2020*

References: The growth equation of cities, Nature

The science of cities is interested in the regularities observed in world's major urban systems. Central to this science is the theoretical understanding of how the population is distributed among cities in a country. Zipf's law, first described for cities in 1913, states that this distribution displays a remarkable regularity: if the cities of a country are ranked according to their population size in descending order, the population of a city is then inversely proportional to its rank. Thus, the most populous city in a country is generally twice as large as the second most populous. A cornerstone of urban geography, Zipf's law shows that urban populations do not tend towards an optimum that would lead to a "single size", but rather that they are very heterogeneous and obey a sort of "hierarchy". This striking regularity has triggered numerous debates and studies for more than one century.

Researchers in economics have suggested that Zipf's law is the result of a random growth process and economic shocks. In particular, Gabaix showed in 1999 that if population growth rates are random and independent of city size, they lead to Zipf's law, as long as the (untested) hypothesis that cities cannot become too small is introduced. So far, this model was considered as the main paradigm for understanding the growth of cities.

However, the dramatic increase of data sources has been a real game-changer. Several recent empirical studies have challenged Zipf's law by identifying numerous variations based on country, time period, or even the definition of cities used for the measurements.

**How cities evolve**

What's more, researchers are not just interested in the distribution of urban populations but also in their evolution over time. In particular, history has shown that cities and civilizations can appear or disappear according to a chaotic dynamic that the Gabaix model is incapable of explaining. Explaining these empirical observations was thus an important challenge for all scientists interested in the modelling of cities.

The IPhT researchers addressed this problem with a new non-deterministic (stochastic) equation, constructed from the empirical analysis of recent data from Canada, France, the United Kingdom and the United States. This equation reveals that it is the rare but very important inter-urban migration shocks that dominate the growth of cities.

The resulting equation predicts a more complex organization of cities than that described by Zipf's law which is therefore not a universal law as it was thought before. This equation produces multiple temporal variations in the hierarchy of cities, in agreement with observations. These results emphasize the importance of rare events in the evolution of complex systems and, at a more practical level the importance of urban planning: it is possible to change the destiny of a city!