Outbreak in the Weighted Random Connection Model

Branding of a new service or a technology is key to business success. Before starting with branding it is important to have a model for the adoption of a service. The model should describe how potential customers react on the offered service, as well as how they influence each other. There are few instrumental elements that we incorporate into the proposed model: risk of the customers (or willingness to accept a new service), the network structure (relationship among the customers), as well as the dynamics that convey the adoption. We propose a new model (the weighted random connection model), where the goal is to express the effect that the probability of adoption of a new service increases in the risk factor, and decays with the distance in the relationship graph of customers. The proposed model incorporates a very general combined risk behavior function for every pair of the customers. This includes many classes of spread, thus can be very applicable on the practical side. Besides the numerical analysis, work can be formally defined and analyzed as an epidemic, that is, as a general dynamical process. Our main result is on the almost sure existence of an infinite component in the weighed random connection model.