This paper formulates a multitask optimization problem where agents in the network have individual objectives to meet, or individual parameter vectors to estimate, subject to a smoothness condition over the graph. The smoothness requirement softens the transition in the tasks among adjacent nodes and allows incorporating information about the graph structure into the solution of the inference problem. A diffusion strategy is devised that responds to streaming data and employs stochastic approximations in place of actual gradient vectors, which are generally unavailable. We show, under conditions on the step-size parameter, that the adaptive strategy induces a contraction mapping and leads to small estimation errors on the order of the small step-size. A graph spectral filtering interpretation is provided for the optimization framework.