Beyond Zipf's Law: The Lavalette Rank Function and its Properties

TL;DR

This paper introduces the Lavalette rank function, an analytical probability distribution derived from the Beta rank function, demonstrating its similarity to the lognormal distribution and its utility in fitting ranked data deviating from Zipf's law.

Contribution

It derives the Lavalette rank function analytically and compares its properties and utility to Zipf's law and the lognormal distribution in data fitting.

Findings

Lavalette distribution is approximately equal to the lognormal distribution.

The Lavalette rank function can effectively fit data deviating from Zipf's law.

Analytical derivation of the Lavalette probability density function.

Abstract

Although Zipf's law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf's law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette rank function, the probability density function can be derived analytically. We also show both computationally and analytically that Lavalette distribution is approximately equal, though not identical, to the lognormal distribution. We illustrate the utility of Lavalette rank function in several datasets. We also address three analysis issues on the statistical testing of Lavalette fitting function, comparison between Zipf's law and lognormal distribution through Lavalette function, and comparison between lognormal distribution and Lavalette distribution.