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.

翻译：Beta秩函数是一种具有平滑峰值和双侧幂律衰减特征的双边分布，广泛用于模拟偏离纯幂律的经验数据。本文提出一种新颖的两步生成过程，能够精确产生符合BRF分布的数据。第一步可采用任意生成幂律分布的机制，第二步则施加回归性再分配过程，将资源从较贫困实体重新分配给较富裕实体，从而加剧不平等性。该方法首次通过生成机制实现了对精确BRF分布的解析推导。我们通过收入与城市人口分布的应用案例验证了该模型。除精确生成外，该框架为理解复杂系统中常见的幂律偏离现象提供了新的系统发生学视角，将秩分布与潜在的反馈机制和再分配动力学联系起来。