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

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.