  Previous work on undirected small-world networks established the paradigm
that locally structured networks tend to have high density of short loops. On
the other hand, many realistic networks are directed. Here we investigate the
local organization of directed networks and find, surprisingly, that real
networks often have very few short loops as compared to random models. We
develop a theory and derive conditions for determining if a given network has
more or less loops than its randomized counterpart. These findings carry broad
implications for structural and dynamical processes sustained by directed
networks.
