Going Viral: Stability of Consensus-Driven Adoptive Spread

TL;DR

This paper models the coevolution of social reinforcement and product adoption in networks, analyzing stability conditions and demonstrating behaviors consistent with sociology findings.

Contribution

It introduces a combined opinion-adoption model with stability analysis, extending to bounded confidence dynamics, and provides insights into the conditions for widespread adoption or rejection.

Findings

Stable all-adopt and none-adopt equilibria identified

Conditions for stability of flop and hit states derived

Model behaviors align with sociology literature on adoption

Abstract

The spread of new products in a networked population is often modeled as an epidemic. However, in the case of `complex' contagion, these models {do not capture nuanced, dynamic social reinforcement effects in adoption behavior}. In this paper, we investigate a model of complex contagion which allows a coevolutionary interplay between adoption, modeled as an SIS epidemic spreading process, and social reinforcement effects, modeled as consensus opinion dynamics. Asymptotic stability analysis of the all-adopt as well as the none-adopt equilibria of the combined opinion-adoption model is provided through the use of Lyapunov arguments. In doing so, sufficient conditions are provided which determine the stability of the `flop' state, where no one adopts the product and everyone's opinion of the product is least favorable, and the `hit' state, where everyone adopts and their opinions are most favorable. These conditions are shown to extend to the bounded confidence opinion dynamic under a stronger assumption on the model parameters. To conclude, numerical simulations demonstrate behavior of the model which reflect findings from the sociology literature on adoption behavior.