We consider networks in which every node updates its value in discrete time by taking a weighted average of the values of the nodes it interacts with. Using an objective function that quantifies the efficiency with which clusters of interacting nodes converge to consensus internally, we formulate an optimization problem that identifies distinct communities in the network. The optimal community detection problem is combinatorial in nature and intractable in general, and we use convex relaxations to reformulate the problem as a semidefinite program. We demonstrate the utility of our algorithm by applying it to some benchmark graphs from the network science literature.