Five degrees of randomness

Abstract

Randomness is omnipresent, and hence the quantification of randomness is a fundamental necessity across the sciences. As “necessity is the mother of invention”, scientists devised various approaches to quantify randomness: statistics uses standard deviation; statistical physics and information theory use entropies (e.g. Shannon); socioeconomics uses inequality indices (e.g. Gini); and ecology uses diversity indices (e.g. Simpson). Alternative to these approaches – which are all continuous quantifications – Mandelbrot suggested a radically different approach: a digital categorization of randomness. Inspired by Mandelbrot, here we showcase a digital categorization comprising five degrees of randomness — á la the Saffir–Simpson hurricane scale, and á la the DEFCON states of defense readiness. Using the reliability-engineering notion of hazard rates, we present a comprehensive study of the digital categorization. From a scholarly viewpoint, we unveil the categorization’s profound connections to Gibbs measures in statistical physics, and to the following probability-theory notions: heavy tails, long tails, slow variation, regular variation, and rapid variation. From an applicative viewpoint, we demonstrate the categorization’s potency and usability. This paper is relevant to wide audiences: theoreticians and practitioners that are tackling random systems and processes.