Power-law distributions from additive preferential redistributions

TL;DR

This paper presents a non-growth, additive preferential redistribution model that produces power-law distributions with the Zipf exponent, applicable to systems like wealth, city sizes, and network rewiring, without relying on growth mechanisms.

Contribution

It introduces a novel non-growth model with additive preferential interactions that generates power-law distributions, expanding understanding beyond traditional growth-based models.

Findings

The model produces stationary power-law distributions analytically and numerically.

Scaling behavior is observed for certain parameter ranges.

The mechanism applies to wealth, city sizes, and network rewiring scenarios.

Abstract

We introduce a non-growth model that generates the power-law distribution with the Zipf exponent. There are N elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule, while the sum of quantities is conserved. The situation described by this model is similar to those of closed $N$-particle systems when conservative two-body collisions are only allowed. We obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model, and find that the model exhibits the scaling behavior for some parameter ranges. Unlike well-known growth models, this alternative mechanism generates the power-law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements. This model can be applied to some examples such as personal wealths, city sizes, and the generation of scale-free networks when only rewiring is allowed.