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 condition 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. The approach relies on minimizing a global cost
consisting of the aggregate sum of individual costs regularized by a term that promotes smoothness. We show
in this Part I of the work, 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. The results in
the accompanying Part II will reveal explicitly the influence of the network topology and the regularization
strength on the network performance and will provide insights into the design of effective multitask strategies
for distributed inference over networks.