The diffusion of an innovation can be represented by a process in which agents choose perturbed best responses to what their neighbors are currently doing. Diffusion is said to be fast if the expected waiting time until the innovation spreads widely is bounded above independently of the size of the network. Previous work has identified specific topological properties of networks that guarantee fast diffusion. Here we apply martingale theory to derive topology-free bounds such that diffusion is fast whenever the payoff gain from the innovation is sufficiently high and the response function is sufficiently noisy. We also provide a simple method for computing an upper bound on the expected waiting time that holds for all networks. For the logit response function, it takes on average less than 80 revisions per capita for the innovation to diffuse widely in any network, when the error rate is at least 5% and the payoff gain (relative to the status quo) is at least 150%. Qualitatively similar results hold for other smoothed best response functions and populations that experience heterogeneous payoff shocks.
