The diffusion of an innovation can be represented by an adaptive 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. One can apply martingale theory to derive bounds such that diffusion is fast whenever the payoff gain from the innovation is sufficiently high and the response function is sufficiently noisy. In fact there is a simple method for computing an upper bound on the expected waiting time that holds for all networks irrespective of their topological characteristics.
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