Continuum rich-get-richer processes: Mean field analysis with an application to firm size

TL;DR

This paper introduces a continuum-based rich-get-richer model using PDEs, generalizes Simon's model with arbitrary growth kernels, extends to multiple dimensions, and applies it to firm size and wealth distributions, matching empirical data.

Contribution

It develops a PDE framework for rich-get-richer processes, generalizes Simon's model, and applies the approach to real-world firm size and wealth data.

Findings

Power law distributions for firm size and wealth.

Model matches empirical and simulation data.

General PDE solutions for multi-dimensional cases.

Abstract

Classical rich-get-richer models have found much success in being able to broadly reproduce the statistics and dynamics of diverse real complex systems. These rich-get-richer models are based on classical urn models and unfold step-by-step in discrete time. Here, we consider a natural variation acting on a temporal continuum in the form of a partial differential equation (PDE). We first show that the continuum version of Herbert Simon's canonical preferential attachment model exhibits an identical size distribution. In relaxing Simon's assumption of a linear growth mechanism, we consider the case of an arbitrary growth kernel and find the general solution to the resultant PDE. We then extend the PDE to multiple spatial dimensions, again determining the general solution. Finally, we apply the model to size and wealth distributions of firms. We obtain power law scaling for both to be concordant with simulations as well as observational data.