Going Viral: Stability of Consensus-Driven Adoptive Spread

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. Additional analysis is provided for the case where the product is neither a hit nor a flop. To conclude, numerical simulations demonstrate behaviors indicated in the sociology literature such as tipping points.