The Bass diffusion equation on scale-free networks with correlations and inhomogeneous advertising
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The Bass diffusion equation, which is an effective forecasting tool for innovation diffusion based on large collections of empirical data, assumes an homogeneous diffusion process. We introduce for the first time a network structure into this model and investigate it numerically in the case of scale-free networks with link density P(k) = c/k^a, where k=1,...,N. The resulting curve of the total adoptions in time is qualitatively similar to the homogeneous Bass curve corresponding to the same average number of connections. The peak of the adoptions, however, tends to occur earlier, particularly when and N are large (i.e., there are few hubs with a large maximum number of connections). Most interestingly, the adoption curve of the hubs anticipates the total adoption curve in a predictable way, with peak times which are, in a typical case with N=15, approx. 60% of the total adoptions peak. This may allow to monitor the hubs for forecasting purposes. We also consider the case of networks with assortative and disassortative correlations, by explicit construction of the relative P(h|k) matrices, and a case of inhomogeneous advertising where the linear terms are "targeted" on the hubs while maintaining their total cost constant.