  A goal in network science is the geometrical characterization of complex
networks. In this direction, we (arXiv:1603.00386; J. Stat. Mech. (2016)
P063206) have recently introduced the Forman's discretization of Ricci
curvature to the realm of undirected networks. Investigation of Forman
curvature in diverse model and real-world undirected networks revealed that
this measure captures several aspects of the organization of complex undirected
networks. However, many important real-world networks are inherently directed
in nature, and the Forman curvature for undirected networks is unsuitable for
analysis of such directed networks. Hence, we here extend the Forman curvature
for undirected networks to the case of directed networks. The simple
mathematical formula for the Forman curvature in directed networks elegantly
incorporates node weights, edge weights and edge direction. By applying the
Forman curvature for directed networks to a variety of model and real-world
directed networks, we show that the measure can be used to characterize the
structure of complex directed networks. Furthermore, our results also hold in
real directed networks which are weighted or spatial in nature. These results
in combination with our previous results suggest that the Forman curvature can
be readily employed to study the organization of both directed and undirected
complex networks.
