Universality of Zipf's Law

TL;DR

This paper demonstrates that Zipf's law naturally emerges from a broad class of stochastic systems under minimal assumptions, explaining its widespread occurrence across various complex systems.

Contribution

It provides a general, model-free theoretical derivation showing Zipf's law as an inevitable outcome of systems balancing order and disorder.

Findings

Zipf's law arises from systems with stable complexity.

The derivation is based on Algorithmic Information Theory.

The result applies broadly to many real-world systems.

Abstract

Zipf's law is the most common statistical distribution displaying scaling behavior. Cities, populations or firms are just examples of this seemingly universal law. Although many different models have been proposed, no general theoretical explanation has been shown to exist for its universality. Here we show that Zipf's law is, in fact, an inevitable outcome of a very general class of stochastic systems. Borrowing concepts from Algorithmic Information Theory, our derivation is based on the properties of the symbolic sequence obtained through successive observations over a system with an unbounded number of possible states. Specifically, we assume that the complexity of the description of the system provided by the sequence of observations is the one expected for a system evolving to a stable state between order and disorder. This result is obtained from a small set of mild, physically relevant assumptions. The general nature of our derivation and its model-free basis would explain the ubiquity of such a law in real systems.