A Rank-Based Approach to Zipf's Law

TL;DR

This paper introduces a rank-based framework using Atlas models to understand when systems follow Zipf's law, providing insights into the dynamics leading to this power-law distribution.

Contribution

It establishes rank-based conditions within Atlas models that result in Zipf's law, linking distributional properties with system dynamics.

Findings

Identifies rank-based criteria for Zipf's law in Atlas models

Connects steady-state power laws with system dynamics

Provides a method to analyze systems exhibiting Zipf's law

Abstract

An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is -1. In this note, rank-based conditions are found under which an Atlas model will follow Zipf's law. An advantage of this rank-based approach is that it provides information about the dynamics of systems that result in Zipf's law.