  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 social influences are controlled by two parameters: $F$, the
number of features that characterizes the cultures and $q$, the common number
of states each feature can assume. 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 among the individuals.
For infinite regular lattices in one (1D) and two dimensions (2D), we find that
initially the successful innovation spreads linearly with the time $t$, but in
the long-time limit it spreads diffusively ($\sim t^{1/2}$) in 1D and
sub-diffusively ($\sim t/\ln t$) 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 $\gamma$ may be a
useful index to characterize the process of diffusion of successful innovations
in diverse scenarios.
