Rank Distributions: Frequency vs. Magnitude

TL;DR

This paper explores the inverse relationship between frequency and magnitude distributions in ranked data, revealing their connection through probability models and nonlinear maps, with implications for understanding power-law and exponential decay patterns.

Contribution

It establishes a theoretical link between frequency and magnitude rank distributions, including their inverse relationship and applications to various decay laws, extending to thermodynamic concepts.

Findings

Identifies the inverse relationship between frequency and magnitude distributions.

Shows that hyperbolic decay distributions are identical (Zipf law).

Highlights differences in decay patterns for various rank distributions.

Abstract

We examine the relationship between two different types of ranked data, frequencies and magnitudes. We consider data that can be sorted out either way, through numbers of occurrences or size of the measures, as it is the case, say, of moon craters, earthquakes, billionaires, etc. We indicate that these two types of distributions are functional inverses of each other, and specify this link, first in terms of the assumed parent probability distribution that generates the data samples, and then in terms of an analog (deterministic) nonlinear iterated map that reproduces them. For the particular case of hyperbolic decay with rank the distributions are identical, that is, the classical Zipf plot, a pure power law. But their difference is largest when one displays logarithmic decay and its counterpart shows the inverse exponential decay, as it is the case of Benford law, or viceversa. For all intermediate decay rates generic differences appear not only between the power-law exponents for the midway rank decline but also for small and large rank. We extend the theoretical framework to include thermodynamic and statistical-mechanical concepts, such as entropies and configuration.