Diffusion of innovations in Axelrod's model

Axelrod's model for the dissemination of culture contains two key factors required to model the process of diffusion of innovations, namely, social influence (i.e. individuals become more similar when they interact) and homophily (i.e. individuals interact preferentially with similar others). The strength of these factors is controlled by two parameters: F, the number of features that characterize the cultures, and q, the common number of states each feature can assume. For fixed F, a large value of q reduces the frequency of interactions between individuals because it makes their cultures more diverse. Here we assume that the innovation is a new state of a cultural feature of a single individual-the innovator-and study how the innovation spreads through the networks of the individuals. For infinite regular lattices in one (1D) and two dimensions (2D), we find that initially successful innovation spreads linearly with time t, but in the long-time limit it spreads diffusively (similar to t(1/2)) in 1D and sub-diffusively (similar to t/ln t) in 2D. For finite lattices, the growth curves for the number of adopters are typically concave functions of t. For random graphs with a finite number of nodes N, we argue that the classical S-shaped growth curves result from a trade-off between the average connectivity K of the graph and the per-feature diversity q. A large q is needed to reduce the pace of the initial spreading of the innovation and thus delimit the early-adopters stage, whereas a large K is necessary to ensure the onset of the take-off stage at which the number of adopters grows superlinearly with t. In an infinite random graph we find that the number of adopters of a successful innovation scales with t(gamma) with gamma = 1 for K > 2 and 1/2 < gamma < 1 for K = 2. We suggest that the exponent. may be a useful index to characterize the process of diffusion of successful innovations in diverse scenarios.