Modeling Two-Scale Rank Distributions via Redistribution Dynamics or an Analytic Derivation of the Beta Rank Function

TL;DR

This paper introduces a novel generative model combining power-law mechanisms and redistribution processes to exactly produce the Beta Rank Function, providing insights into systemic inequality and deviations from pure power laws.

Contribution

It presents the first analytic derivation of the Beta Rank Function from a two-step generative process involving power-law generation and redistribution.

Findings

Validates the model with income and urban population data.

Links rank distribution deviations to feedback and redistribution dynamics.

Provides a new theoretical framework for systemic inequality analysis.

Abstract

Beta Rank Function (BRF) is a two-sided distribution characterized by a smooth peak and double powerlaw decay, widely used to model empirical data exhibiting deviations from pure power laws. In this paper, we introduce a novel two-step generative process that produces data exactly following the BRF distribution. The first step involves any mechanism generating a power-law distribution, while the second step applies a regressive redistribution process that reallocates resources from poorer to richer entities, thereby amplifying inequality. This approach represents the first analytic derivation of an exact BRF distribution from a generative mechanism. We validate the model through applications to income and urban population distributions. Beyond exact generation, this framework offers new insights into the systemic origins of deviations from power laws frequently observed in complex systems, linking rank distributions to underlying feedback and redistribution dynamics.