Marketing new products: Bass models on random graphs

Abstract

We consider the problem of marketing a new product in a population modelled as a random graph, in which each individual (node) has a random number of connections to other individuals. Marketing can occur via word of mouth along edges, or via advertising. Our main result is an adaptation of the Miller-Volz model, describing the spread of an infectious disease, to this setting, leading to a generalized Bass marketing model. The Miller-Volz model can be directly applied to word-of-mouth marketing. The main challenge lies in revising the Miller-Volz model to incorporate advertisement, which we solve by introducing a marketing node that is connected to every individual in the population. We tested this model for Poisson and scale-free random networks, and found excellent agreement with microscopic simulations. In the homogeneous limit where the number of individuals goes to ∞ and the network is completely connected our model becomes the classical Bass model. We further present the generalization of this model to two competing products. For a completely connected network this model is again consistent with the known continuum limit. Numerical simulations show excellent agreement with microscopic simulations obtained via an adaptation of the Gillespie algorithm. Our model shows that, if the two products have the same word-of-mouth marketing rate on the network, then the ratio of their market shares is exactly the ratio of their advertisement rates.