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

Oscar Fontanelli, Wentian Li

TL;DR

The paper addresses why rank-size data often deviate from pure power laws by introducing the Beta Rank Function (BRF), a two-scale model with a central peak and distinct left and right tails. It derives an exact analytic generative mechanism: starting from a Pareto distribution and applying a regressive redistribution via the quantile transform $Y=cX[F_X(X)]^b$, the BRF with quantile $Q_{BRF}(p)= C \frac{p^b}{(1-p)^a}$ is obtained. Validation is provided through numerical simulations and empirical fits to income and urban-population data, illustrating the emergence of two characteristic scales governed by $a$ and $b$. Overall, the work links redistribution dynamics to two-scale rank distributions and offers a rigorous, interpretable tool for modeling complex systems with elbows in log-log rank-size plots.

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.

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

TL;DR

The paper addresses why rank-size data often deviate from pure power laws by introducing the Beta Rank Function (BRF), a two-scale model with a central peak and distinct left and right tails. It derives an exact analytic generative mechanism: starting from a Pareto distribution and applying a regressive redistribution via the quantile transform , the BRF with quantile is obtained. Validation is provided through numerical simulations and empirical fits to income and urban-population data, illustrating the emergence of two characteristic scales governed by and . Overall, the work links redistribution dynamics to two-scale rank distributions and offers a rigorous, interpretable tool for modeling complex systems with elbows in log-log rank-size plots.

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.
Paper Structure (11 sections, 1 theorem, 10 equations, 5 figures)

This paper contains 11 sections, 1 theorem, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Some previously proposed generating mechanisms for the DGBD rank function/BRF distribution
  4. Basic aspects of quantile function modeling
  5. Results
  6. Modulated Pareto distribution yields a BRF
  7. Numerical simulations
  8. Example: income distribution
  9. Example: urban area population distribution
  10. Discussion
  11. Conclusions

Key Result

Proposition 1

Let $X \sim \hbox{Pareto}(x_m,a)$. The random variable $Y$ defined by multiplying $X$ with its cumulative density function to the $b$-th power ($b>0$) follows a BRF distribution with parameters $x_m$, $a$ and $b$.

Figures (5)

  • Figure 1: Quantile - rank size function correspondence. Top panels show the probability density and the quantile function of a lognormal distribution with standard parameters. Bottom panels show the histogram and rank-size plot of a random sample (size=100) from the same distribution.
  • Figure 2: A) Power law contraction. A progressive contraction transforms a power law into a DGBD/BRF distribution. With proper normalization, this transformation can be interpreted as reallocation of resources from the smallest to the largest entities. B) Quantile functions for power distribution, Pareto and BRF. Quantile functions of a Pareto distribution $Q_{pareto}(p)=0.3/(1-p)^{0.9}$ (black solid line), a power-law distribution $Q_{power-law}(p)=p^{0.6}$ (black dashed line), their product $Q_{BRF}(p)=0.3 p^{0.6}/(1-p)^{0.9}$ (red solid line), and another product with an extra coefficient of $2^b$: $Q_{BRF}(p)=0.3 (2p)^{0.6}/(1-p)^{0.9}$ (pink solid line). The quantile values at p=0.4 are marked for the Pareto distribution (black dot), the power-law modified(circle), and their product (red dot). The median (quantile value at p=0.5) is marked for the Pareto distribution and the second BRF distribution (the two medians are the same). This figure is modified from the Fig.1.12 in li1992random.
  • Figure 3: Effect of power law contraction. On each column we show the shape of the modulation function with different values of $b$ (lower panels), as well as the resulting distributions, both on log-histogram (upper panels) and rank-size (medium panels) representation. All these are results from stochastic, numerical simulations.
  • Figure 4: Modelling the power law transformation in income distribution. On the left side we show simulated pure-power laws and how they transform into unimodal distributions after modulation or redistribution. On the right side we show the log-histogram of income data for Italy and Mexico, as well as observed rank-size plot and fitted BRF (in red).
  • Figure 5: Modelling the power law transformation in population distribution. On the left side we show a simulated pure-power law and how it transforms into a unimodal distribution after modulation or redistribution. On the right side we show the log-histogram of urban area population in the US, as well as observed rank-size plot and fitted BRF (in red).

Theorems & Definitions (2)

  • Proposition 1
  • proof