Aggregate diffusion dynamics in agent-based models with a spatial structure

We explicitly calculate the aggregate diffusion dynamics in one-dimensional agent-based models of adoption of new products, without using the mean-field approximation. We then introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics in multidimensional agent-based models. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with "any" spatial structure.