Marketing new Products : 1 Bass Models on Random Graphs

5 We consider the problem of marketing a new product in a population 6 modelled as a random graph, in which each individual (node) has a 7 random number of connections to other individuals. Marketing can 8 occur via word of mouth along edges, or via advertising. Our main 9 result is an adaptation of the Miller-Volz model, describing the spread 10 of an infectious disease, to this setting, leading to a generalized Bass 11 marketing model. The Miller-Volz model can be directly applied to 12 word-of-mouth marketing. The main challenge lies in revising the 13 Miller-Volz model to incorporate advertisement, which we solve by 14 introducing a marketing node that is connected to every individual 15 in the population. We tested this model for Poisson and scale free 16 random networks, and found excellent agreement with microscopic 17 simulations. In the homogeneous limit where the number of individ18 uals goes to ∞ and the network is completely connected our model 19 becomes the classical Bass model. We further present the general20 ization of this model to two competing products. For a completely 21 connected network this model is again consistent with the known con22 tinuum limit. Numerical simulations show excellent agreement with 23