Link prediction and clustering are key problems for network-structureddata. While spectral clustering has strong theoretical guaranteesunder the popular stochastic blockmodel formulation of networks, itcan be expensive for large graphs. On the other hand, the heuristic ofpredicting links to nodes that share the most common neighbors withthe query node is much fast, and works very well in practice. We showtheoretically that the common neighbors heuristic can extract clustersw.h.p. when the graph is dense enough, and can do so even in sparsergraphs with the addition of a ``cleaning'' step. Empirical results onsimulated and real-world data support our conclusions.