Geometric description of clustering in directed networks

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.