Zipf's law and city sizes: A short tutorial review on multiplicative processes in urban growth

TL;DR

This paper reviews how multiplicative stochastic processes explain the power-law distribution of city sizes, focusing on Zipf's law and Simon's model, highlighting its flexibility and open problems in urban growth modeling.

Contribution

It provides an accessible tutorial on Simon's model for city size distributions, emphasizing its extensions and discussing open challenges in predicting Zipf's rank plots.

Findings

Simon’s model explains Zipf's law in city sizes.

The model's flexibility allows for various extensions.

Open problems remain in accurately predicting rank plot shapes.

Abstract

We address the role of multiplicative stochastic processes in modeling the occurrence of power-law city size distributions. As an explanation of the result of Zipf's rank analysis, Simon's model is presented in a mathematically elementary way, with a thorough discussion of the involved hypotheses. Emphasis is put on the flexibility of the model, as to its possible extensions and the relaxation of some strong assumptions. We point out some open problems regarding the prediction of the detailed shape of Zipf's rank plots, which may be tackled by means of such extensions.