A universal rank-size law

TL;DR

This paper introduces a universal rank-size law derived from physics principles, offering a more accurate alternative to Zipf's law for describing rank-size relationships across various domains.

Contribution

It proposes a new theoretical distribution based on the Yule-Simon distribution and entropy arguments, providing a universal form for rank-size relationships.

Findings

The new law better fits empirical rank-size data than Zipf's law.

Illustrations include city rankings and sports competition rankings.

Theoretical foundation suggests an optimal distribution based on entropy.

Abstract

A mere hyperbolic law, like the Zipf's law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the "best (or optimal) distribution", is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations.