  Many products and innovations become well-known and widely adopted through
the social interactions of individuals in a population. The Bass diffusion
model has been widely used to model the temporal evolution of adoption in such
social systems. In the model, the likelihood of a new adoption is proportional
to the number of previous adopters, implicitly assuming a global (or
homogeneous) interaction among all individuals in the network. Such global
interactions do not exist in many large social networks, however. Instead,
individuals typically interact with a small part of the larger population. To
quantify the growth rate (or equivalently the adoption timing) in networks with
limited interactions, we study a stochastic adoption process where the
likelihood that each individual adopts is proportional to the number of
adopters among the small group of persons he/she interacts with (and not the
entire population of adopters). When the underlying network of interactions is
a random $k$-regular graph, we compute the sample path limit of the fraction of
adopters. We show the limit coincides with the solution of a differential
equation which can viewed as a generalization of the Bass diffusion model. When
the degree $k$ is bounded, we show the adoption curve differs significantly
from the one corresponds to the Bass diffusion model. In particular, the
adoption grows more slowly than what the Bass model projects. In addition, the
adoption curve is asymmetric, unlike that of the Bass diffusion model. Such
asymmetry has important consequences for the estimation of market potential.
Finally, we calculate the timing of early adoptions at finer scales, e.g.,
logarithmic in the population size.
