The Bass diffusion model on finite Barabasi-Albert networks

TL;DR

This paper analyzes the diffusion times of the Bass model on finite Barabasi-Albert networks, revealing minimal diffusion times due to their unique degree correlation properties and how initial stimuli on hubs affect diffusion speed.

Contribution

It provides an exact mean-field formulation of the Bass model on BA networks and compares diffusion times with other scale-free networks, highlighting the impact of degree correlations.

Findings

Diffusion times are minimized on finite BA networks.

Degree correlations in BA networks influence diffusion speed.

Enhanced initial stimuli on hubs increase diffusion speed differences.

Abstract

Using a mean-field network formulation of the Bass innovation diffusion model and exact results by Fotouhi and Rabbat on the degree correlations of Barabasi-Albert networks, we compute the times of the diffusion peak and compare them with those on scale-free networks which have the same scale-free exponent but different assortativity properties. We compare our results with those obtained by Caldarelli et al. for the SIS epidemic model with the spectral method applied to adjacency matrices. It turns out that diffusion times on finite Barabasi-Albert networks are at a minimum. This may be due to a little-known property of these networks: although the value of the assortativity coefficient is close to zero, they look disassortative if one considers only a bounded range of degrees, including the smallest ones, and slightly assortative on the range of the higher degrees. We also find that if the trickle-down character of the diffusion process is enhanced by a larger initial stimulus on the hubs (via a inhomogeneous linear term in the Bass model), the relative difference between the diffusion times for BA networks and uncorrelated networks is even larger, reaching for instance the 34% in a typical case on a network with $10^4$ nodes.