Tail universalities in rank distributions as an algebraic problem: the beta-like function

TL;DR

This paper introduces a beta-like function that universally fits rank distributions across various fields, addressing tail-fitting issues of traditional power-law models by linking the distribution to algebraic properties of product of variables.

Contribution

It proposes a new beta-like distribution model for rank data, grounded in algebraic analysis of product of variables, improving tail fit over existing models.

Findings

Beta-like function fits diverse rank data well

Universal behavior explained by product of subsystems

Algebraic approach links distribution to system composition

Abstract

Although power laws of the Zipf type have been used by many workers to fit rank distributions in different fields like in economy, geophysics, genetics, soft-matter, networks etc., these fits usually fail at the tails. Some distributions have been proposed to solve the problem, but unfortunately they do not fit at the same time both ending tails. We show that many different data in rank laws, like in granular materials, codons, author impact in scientific journal, etc. are very well fitted by a beta-like function. Then we propose that such universality is due to the fact that a system made from many subsystems or choices, imply stretched exponential frequency-rank functions which qualitatively and quantitatively can be fitted with the proposed beta-like function distribution in the limit of many random variables. We prove this by transforming the problem into an algebraic one: finding the rank of successive products of a given set of numbers.