Compartmental limit of discrete Bass models on networks

TL;DR

This paper introduces a new method to analyze the convergence of discrete Bass models on networks to their compartmental counterparts, providing explicit rates of convergence and insights into the effects of network structure and heterogeneity.

Contribution

A novel method for proving convergence and rates of convergence of discrete Bass models to compartmental models on various networks.

Findings

Complete homogeneous networks converge at rate 1/M.

Circular networks have exponential convergence rates.

Heterogeneity among groups slows diffusion when influences are positively related.

Abstract

We introduce a new method for proving the convergence and the rate of convergence of discrete Bass models on various networks to their respective compartmental Bass models, as the population size $M$ becomes infinite. In this method, the full set of master equations is reduced to a smaller system of equations, which is closed and exact. The reduced finite system is embedded into an infinite system, and the convergence of that system to the infinite limit system is proved using standard ODE estimates. Finally, an ansatz provides an exact closure of the infinite limit system, which reduces that system to the compartmental model. Using this method, we show that when the network is complete and homogeneous, the discrete Bass model converges to the original 1969 compartmental Bass model, at the rate of $1/M$. When the network is circular, however, the compartmental limit is different, and the rate of convergence is exponential in $M$. In the case of a heterogeneous network that consists of $K$ homogeneous groups, the limit is given by a heterogeneous compartmental Bass model, and the rate of convergence is $1/M$. Using this compartmental model, we show that when the heterogeneity in the external and internal influence parameters among the $K$ groups is positively monotonically related, heterogeneity slows down the diffusion.