It has often been taken as a working assumption that directed links in information networks are frequently formed by "short-cutting'' a two-step path between the source and the destination — a kind of implicit "link copying'' analogous to the process of triadic closure in social networks. Despite the role of this assumption in theoretical models such as preferential attachment, it has received very little direct empirical investigation. Here we develop a formalization and methodology for studying this type of directed closure process, and we provide evidence for its important role in the formation of links on Twitter. We then analyze a sequence of models designed to capture the structural phenomena related to directed closure that we observe in the Twitter data.