We derive an adaptive diffusion mechanism to optimize global cost functions in a distributed manner over a network of nodes. The cost function is assumed to consist of the sum of individual components, and diffusion adaptation is used to enable the nodes to cooperate locally through in-network processing in order to solve the desired optimization problem. We analyze the mean-square-error performance of the algorithm, including its transient and steady-state behavior. We illustrate one application in the context of least-mean-squares estimation for sparse vectors.