Models of Universal Power-Law Distributions

TL;DR

This paper introduces a simple, robust model that explains the emergence of Zipf's law and other power-law distributions through asymmetric random walks, with extensions accounting for various exponents.

Contribution

The paper presents a generic model that reproduces Zipf's law and extends to various power-law exponents, supported by theoretical analysis.

Findings

Model reproduces Zipf's law robustly.

Extended model captures a range of power-law exponents.

Theoretical explanation links Zipf's law to asymmetric random walks.

Abstract

Power-law distributions with various exponents are studied. We first introduce a simple and generic model that reproduces Zipf's law. We can regard this model both as the time evolution of the population of cities and that of the asset distribution. We show that our model is very robust against various variations. Next, we explain theoretically why our model reproduces Zipf's law. By considering the time-evolution equation of our model, we see that the essence of Zipf's law is an asymmetric random walk in a logarithmic scale. Finally, we extend our model by introducing an additional asymmetry. We show that the extended model reproduces various power-law exponents. By extending the theoretical argument for Zipf's law, we find a simple equation of the power-law exponent.