Boundary Effects in the Discrete Bass Model

TL;DR

This paper introduces new analytical tools to study how boundaries affect product diffusion in networks, revealing that one-sided influence slows diffusion compared to two-sided influence on finite lines.

Contribution

The paper presents the indifference and dominance principles, providing explicit methods to analyze and compare diffusion processes on different network structures.

Findings

One-sided diffusion is slower than two-sided diffusion on finite lines.

Diffusion on a circle shows no difference between one-sided and two-sided influence.

Numerical evidence suggests similar boundary effects in higher dimensions.

Abstract

To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.