Phase transition in the rich-get-richer mechanism due to finite-size effects

TL;DR

This paper investigates how finite-size effects induce a phase transition in wealth distribution under the rich-get-richer mechanism, revealing different asymptotic behaviors depending on the fraction of wealth-increasing steps.

Contribution

It introduces analytical and numerical analysis of the rich-get-richer process with a finite number of agents, uncovering a phase transition in the resulting wealth distribution.

Findings

For r < 1/2, the distribution approaches a Gaussian form.

For r > 1/2, the distribution exhibits a Pareto tail and stretched exponential decay.

Finite-size effects cause a phase transition in the wealth distribution.

Abstract

The rich-get-richer mechanism (agents increase their ``wealth'' randomly at a rate proportional to their holdings) is often invoked to explain the Pareto power-law distribution observed in many physical situations, such as the degree distribution of growing scale free nets. We use two different analytical approaches, as well as numerical simulations, to study the case where the number of agents is fixed and finite (but large), and the rich-get-richer mechanism is invoked a fraction r of the time (the remainder of the time wealth is disbursed by a homogeneous process). At short times, we recover the Pareto law observed for an unbounded number of agents. In later times, the (moving) distribution can be scaled to reveal a phase transition with a Gaussian asymptotic form for r < 1/2 and a Pareto-like tail (on the positive side) and a novel stretched exponential decay (on the negative side) for r > 1/2.