A model for the size distribution of customer groups and businesses

We present a generalization of the dynamical model of information transmission and herd behavior proposed by Eguiluz and Zimmermann. A characteristic size of group of agents $s_{0}$ is introduced. The fragmentation and coagulation rates of groups of agents are assumed to depend on the size of the group. We present results of numerical simulations and mean field analysis. It is found that the size distribution of groups of agents $n_{s}$ exhibits two distinct scaling behavior depending on $s \leq s_{0}$ or $s > s_{0}$. For $s \leq s_{0}$, $n_{s} \sim s^{-(5/2 + \delta)}$, while for $s > s_{0}$, $n_{s} \sim s^{-(5/2 -\delta)}$, where $\delta$ is a model parameter representing the sensitivity of the fragmentation and coagulation rates to the size of the group. Our model thus gives a tunable exponent for the size distribution together with two scaling regimes separated by a characteristic size $s_{0}$. Suitably interpreted, our model can be used to represent the formation of groups of customers for certain products produced by manufacturers. This, in turn, leads to a distribution in the size of businesses. The characteristic size $s_{0}$, in this context, represents the size of a business for which the customer group becomes too large to be kept happy but too small for the business to become a brand name.