Funnel Theorems for Spreading on Networks

Abstract: We derive novel analytic tools for the discrete Bass model, which models the diffusion of new products on networks. We prove that the probability that any two nodes adopt by time t, is greater than or equal to the product of the probabilities that each of the two nodes adopts by time t. We introduce the notion of an "influential node", and use it to determine whether the above inequality is strict or an equality. We then use the above inequality to prove the "funnel inequality", which relates the adoption probability of a node to the product of its adoption probability on two sub-networks. We introduce the notion of a "funnel node", and use it to determine whether the funnel inequality is strict or an equality. The above analytic tools can be exptended to epidemiological models on networks. We then use the funnel theorems to derive a new inequality for diffusion on circles and a new explicit expression for the adoption probabilities of nodes on two-sided line, and to prove that the adoption level on one-sided lines is strictly slower than on anisotropic two-sided lines, and that the adoption level on multi-dimensional Cartesian networks is bounded from below by that on one-dimensional networks.