  Recent advances in the study of networked systems have highlighted that our
interconnected world is composed of networks that are coupled to each other
through different "layers" that each represent one of many possible subsystems
or types of interactions. Nevertheless, it is traditional to aggregate
multilayer networks into a single weighted network in order to take advantage
of existing tools. This is admittedly convenient, but it is also extremely
problematic, as important information can be lost as a result. It is therefore
important to develop multilayer generalizations of network concepts. In this
paper, we analyze triadic relations and generalize the idea of transitivity to
multiplex networks. By focusing on triadic relations, which yield the simplest
type of transitivity, we generalize the concept and computation of clustering
coefficients to multiplex networks. We show how the layered structure of such
networks introduces a new degree of freedom that has a fundamental effect on
transitivity. We compute multiplex clustering coefficients for several real
multiplex networks and illustrate why one must take great care when
generalizing standard network concepts to multiplex networks. We also derive
analytical expressions for our clustering coefficients for ensemble averages of
networks in a family of random multiplex networks. Our analysis illustrates
that social networks have a strong tendency to promote redundancy by closing
triads at every layer and that they thereby have a different type of multiplex
transitivity from transportation networks, which do not exhibit such a
tendency. These insights are invisible if one only studies aggregated networks.
