Models of Social Contagion

During our earliest looks at word-of-mouth interactions, aimed at things like advertising, we examined the structure of customer interaction. Also that context drives this kind of interaction.   This overall is a very technical paper,  but it has some embedded gems.   Explanatory diagrams at the link.

Simplicial models of social contagion
Iacopo Iacopini, Giovanni Petri, Alain Barrat & Vito Latora 
Nature Communications 10, Article number: 2485 (2019) | Download Citation

Abstract
Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

Introduction
Complex networks describe well the connectivity of systems of various nature1,2 and are widely used as the underlying—and possibly multilayered3—social structure on which dynamical processes4,5, such as disease spreading6, diffusion and adoption of innovation7,8,9, and opinion formation10 occur. For example, when modeling an epidemic spreading in a population6, the transmission between infectious and healthy individuals is typically assumed: (i) to occur through pairwise interactions between infectious and healthy individuals, and (ii) to be caused even by a single exposure of a healthy individual to an infectious one. Such processes of simple contagion can be conveniently represented by transmission mechanisms along the links of the network of contacts between individuals.

When dealing instead with social contagion phenomena, such as the adoption of norms, behaviors or new products, or the diffusion of rumors or fads, the situation is more complex. Simple epidemic-like contagion can suffice to describe some cases, such as easily convincing rumors or domino effects11. In other situations, however, they do not provide a satisfactory description, especially in those cases where more complex dynamics of peer influence and reinforcement mechanisms are at work12. Complex contagion mechanisms have been proposed to account for these effects. As defined by Centola and Macy11: “a contagion is complex if its transmission requires an individual to have contact with two or more sources of activation”, i.e. if a “contact with a single active neighbor is not enough to trigger adoption”. Complex contagion can hence be broadly defined as a process in which exposure to multiple sources presenting the same stimulus is needed for the contagion to occur. Empirical evidence that contagion processes including multiple exposure can be needed to describe social contagion has been provided in various contexts and experiments13,14,15,16,17.  ... '