  Many social, biological, and economic systems can be approached by complex
networks of interacting units. The behaviour of several models on small-world
networks has recently been studied. These models are expected to capture the
essential features of the complex processes taking place on real networks like
disease spreading, formation of public opinion, distribution of wealth, etc. In
many of these systems relations are directed, in the sense that links only act
in one direction (outwards or inwards). We investigate the effect of directed
links on the behaviour of a simple spin-like model evolving on a small-world
network. We show that directed networks may lead to a highly nontrivial phase
diagram including first and second-order phase transitions out of equilibrium.
