A Generalized Bass Model for Product Growth in Networks

Many products and innovations become well-known and widely adopted through the social interactions of individuals in a population. The Bass diffusion model has been widely used to model the temporal evolution of adoption in such social systems. In the model, the likelihood of a new adoption is proportional to the number of previous adopters, implicitly assuming a global (or homogeneous) interaction among all individuals in the network. Such global interactions do not exist in many large social networks, however. Instead, individuals typically interact with a small part of the larger population. To quantify the growth rate (or equivalently the adoption timing) in networks with limited interactions, we study a stochastic adoption process where the likelihood that each individual adopts is proportional to the number of adopters among the small group of persons he/she interacts with (and not the entire population of adopters). When the underlying network of interactions is a random $k$-regular graph, we compute the sample path limit of the fraction of adopters. We show the limit coincides with the solution of a differential equation which can viewed as a generalization of the Bass diffusion model. When the degree $k$ is bounded, we show the adoption curve differs significantly from the one corresponds to the Bass diffusion model. In particular, the adoption grows more slowly than what the Bass model projects. In addition, the adoption curve is asymmetric, unlike that of the Bass diffusion model. Such asymmetry has important consequences for the estimation of market potential. Finally, we calculate the timing of early adoptions at finer scales, e.g., logarithmic in the population size.