Geometric description of clustering in directed networks

First-principle network models are crucial to understanding the intricate topology of real complex networks. Although modelling efforts have been quite successful in undirected networks, generative models for networks with asymmetric interactions are still not well developed and unable to reproduce several basic topological properties. Progress in this direction is of particular interest, as real directed networks are the norm rather than the exception in many natural and human-made complex systems. Here we show how the network geometry paradigm can be extended to the case of directed networks. We define a maximum entropy ensemble of random geometric directed graphs with a given sequence of in-degrees and out-degrees. Beyond these local properties, the ensemble requires only two additional parameters to fix the levels of reciprocity and the frequency of the seven possible types of three-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 directed complex networks.