Universal emergence of local Zipf-Mandelbrot law

TL;DR

This paper demonstrates that the Zipf-Mandelbrot law naturally emerges in small samples of i.i.d. data, providing a universal explanation for its widespread occurrence across various phenomena.

Contribution

The authors derive an analytical expression for the cumulants of ranked i.i.d. samples and prove that small subsets follow the Zipf-Mandelbrot law.

Findings

Small samples of i.i.d. data exhibit Zipf-Mandelbrot distribution.

Analytical cumulant expressions support the universality of ZM law.

Empirical examples validate theoretical predictions.

Abstract

A plethora of natural and socio-economic phenomena share a striking statistical regularity, that is the magnitude of elements decreases with a power law as a function of their position in a ranking of magnitude. Such regularity is known as Zipf-Mandelbrot law (ZM), and plenty of problem-specific explanations for its emergence have been provided in different fields. Yet, an explanation for ZM ubiquity is currently lacking. In this paper we first provide an analytical expression for the cumulants of any ranked sample of i.i.d. random variables once sorted in decreasing order. Then we make use of this result to rigorously demonstrate that, whenever a small fraction of such ranked dataset is considered, it becomes statistically indistinguishable from a ZM law. We finally validate our results against several relevant examples.