This letter proposes a general regularization framework for inference over multitask networks. The optimization approach relies on minimizing a global cost consisting of the aggregate sum of individual costs regularized by a term that allows to incorporate global information about the graph structure and the individual parameter vectors into the solution of the inference problem. An adaptive strategy, which responds to streaming data and employs stochastic approximations in place of actual gradient vectors, is devised and studied. Methods allowing the distributed implementation of the regularization step are also discussed. This letter shows how to blend real-time adaptation with graph filtering and a generalized regularization framework to result in a graph diffusion strategy for distributed learning over multitask networks.