Zipf's Law for Atlas Models

TL;DR

This paper provides a mathematical explanation for Zipf's law in ranked data using Atlas models, showing it holds under specific conditions and explaining its universality across various systems.

Contribution

It introduces conditions under which Atlas models produce Zipf's law, clarifying the universality and limitations of Zipfian distributions in ranked data.

Findings

Zipf's law arises in Atlas models under conservation and completeness conditions.

Ranked data from certain systems follow Zipf's law, explained by the model.

Some systems follow non-Zipfian Pareto distributions, not explained by the model.

Abstract

A set of data with positive values follows a Pareto distribution if the log-log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf's law if the log-log plot has a slope of -1. Since many types of ranked data follow Zipf's law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of positive continuous semimartingales with parameters that depend only on rank. We show that the stable distribution of an Atlas model will follow Zipf's law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf's law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf's law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, the intensity of solar flares, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.