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The diffusion of an innovation such as a new communication service is modeled using a randomly generated network of interpersonal influence between members of a social system. Each member decides to adopt the innovation when K out of his particular set of L influencers have already become adopters. A discrete-time recursion is obtained which describes the evolution in time of the fraction of adopters. The model exhibits a critical mass effect, where the number of initial adopters must exceed a critical value for the diffusion to saturate the population, otherwise the diffusion ends prematurely without substantial growth. Also exhibited is a bandwagon effect, which explains a sudden upsurge in growth following an initial period of slow diffusion. Both phenomena are amply corroborated by Monte Carlo simulations for a population of 10 000 people.