Underestimating extreme events in power-law behavior due to machine-dependent cutoffs

TL;DR

This paper reveals that standard algorithms for generating power-law distributed random variates are limited by machine-dependent cutoffs, leading to deterministic tail sampling and finite moments, which can distort scientific analyses.

Contribution

It identifies and quantifies machine-dependent limitations in standard power-law sampling algorithms, highlighting their impact on statistical properties.

Findings

Sampling in the tail becomes deterministic due to machine precision limits.

Sample moments converge to finite values instead of diverging as expected.

Standard libraries may not reliably handle certain power-law parameters.

Abstract

Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law distributions. The task is usually performed through standard methods that map uniform random variates into the desired probability space. Whereas all these algorithms are theoretically solid, in this paper we show that they are subject to severe machine-dependent limitations. As a result, two dramatic consequences arise: (i) the sampling in the tail of the distribution is not random but deterministic; (ii) the moments of the sample distribution, which are theoretically expected to diverge as functions of the sample sizes, converge instead to finite values. We provide quantitative indications for the range of distribution parameters that can be safely handled by standard libraries used in computational analyses. Whereas our findings indicate possible reinterpretations of numerical results obtained through flawed sampling methodologies, they also pave the way for the search for a concrete solution to this central issue shared by all quantitative sciences dealing with complexity.