A fundamental problem in the analysis of network data is the detection of network communities, groups of densely interconnected nodes, which may be overlapping or disjoint. Here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models. We show how the method can be implemented using a fast, closed-form expectation-maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times. We test the method both on real-world networks and on synthetic benchmarks and find that it gives results competitive with previous methods. We also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method, and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem.