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

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 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. We then relax the zero-diffusion assumption and find an envelope of solutions to the general model in the presence of small fluctuations. 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, providing a parsimonious theoretical explanation for these phenomena.

Figure 1

Solutions to Eq. (4) with growth kernel $r\left(x\right)=x$ and innovation rate $\rho \left(t\right)={\rho }_{\infty }$.

Figure 2

Solutions to Eq. (4) with growth kernel $r\left(x\right)=x^{\eta }$ for $\eta >1$ and innovation rate $\rho \left(t\right)={\rho }_{\infty }$.

Figure 3

Power-law size distribution of firms. Distribution of firm sizes by employment from 1977 to 2013, obtained from the U.S. Census Bureau on 28 August 2016 [12]. Note that U.S. firms exhibit constant returns to scale, so this implies a power-law distribution (with identical exponent) in firm income. This result was famously publicized by Axtell in Ref. [13]. The wide binning in the figure is due to the lack of granularity in publicly available U.S. Census data on firms with more than ${10}^4$ employees. The inset displays ${\langle f(x,t)\rangle }_x$ over the entire date range. These data exhibit linear scaling, as predicted by Eq. (34).

Figure 4

The fraction of wealth belonging to firms with customer bases greater than or equal to $x$ for $\rho =0.5$, 0.1, and 0.01. We set the price level $p=2.5$ for ease of visualization.