  First principle network models are crucial to make sense of the intricate
topology of real complex networks. While modeling efforts have been quite
successful in undirected networks, generative models for networks with
asymmetric interactions are still not well developed and are unable to
reproduce several basic topological properties. This is particularly
disconcerting considering that real directed networks are the norm rather than
the exception in many natural and human-made complex systems. In this paper, we
fill this gap and show how the network geometry paradigm can be elegantly
extended to the case of directed networks. We define a maximum entropy ensemble
of geometric (directed) random graphs with a given sequence of in- and
out-degrees. Beyond these local properties, the ensemble requires only two
additional parameters to fix the level of reciprocity and the seven possible
types of 3-node cycles in directed networks. A systematic comparison with
several representative empirical datasets shows that fixing the level of
reciprocity alongside the coupling with an underlying geometry is able to
reproduce the wide diversity of clustering patterns observed in real complex
directed networks.
