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

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.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of rank‑size distributions: while the Discrete Generalized Beta Distribution (DGBD) and its associated Beta Rank Function (BRF) have been widely used to model empirical data that deviate from pure power laws, no exact generative mechanism for the BRF has been proposed. The authors introduce a simple two‑step stochastic construction that yields data following the BRF distribution exactly.

First, a set of N observations is drawn from a Pareto (power‑law) distribution X∼Pareto(xₘ, a). The second step applies a “regressive redistribution” transformation: each observation is multiplied by the b‑th power of its own cumulative probability, i.e., Y = X·