  Spreading (diffusion) of innovations is a stochastic process on social
networks. When the key driving mechanism is peer effects (word of mouth), the
rate at which the aggregate adoption level increases with time depends strongly
on the network structure. In many applications, however, the network structure
is unknown. To estimate the aggregate adoption level for such innovations, we
show that the two networks that correspond to the slowest and fastest adoption
levels are a homogeneous two-node network and a homogeneous infinite complete
network, respectively. Solving the stochastic Bass model on these two networks
yields explicit lower and upper bounds for the adoption level on any network.
These bounds are tight, and they also hold for the individual adoption
probabilities of nodes. The gap between the lower and upper bounds increases
monotonically with the ratio of the rates of internal and external influences.
