Universal bounds for spreading on networks

Spreading (diffusion) of innovations is a stochastic process on social networks. When the key driving mechanism is the peer effect (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 as a function of time for such innovations, we show that the minimal and maximal adoption levels are attained on a homogeneous two-node network and on a homogeneous infinite complete network, respectively. Solving the Bass model on these two networks yields explicit lower and upper bounds for the expected 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.