  A goal in network science is the geometrical characterization of complex
networks. In this direction, we have recently introduced Forman's
discretization of Ricci curvature to the realm of undirected networks.
Investigation of this edge-centric network measure, Forman-Ricci curvature, in
diverse model and real-world undirected networks revealed that the curvature
measure captures several aspects of the organization of undirected complex
networks. However, many important real-world networks are inherently directed
in nature, and the definition of the Forman-Ricci curvature for undirected
networks is unsuitable for the analysis of such directed networks. Hence, we
here extend the Forman-Ricci curvature for undirected networks to the case of
directed networks. The simple mathematical formula for the Forman-Ricci
curvature of a directed edge elegantly incorporates vertex weights, edge
weights and edge direction. Furthermore we have compared the Forman-Ricci
curvature with the adaptation to directed networks of another discrete notion
of Ricci curvature, namely, the well established Ollivier-Ricci curvature.
However, the two above-mentioned curvature measures do not account for
higher-order correlations between vertices. To this end, we adjusted Forman's
original definition of Ricci curvature to account for directed simplicial
complexes and also explored the potential of this new, augmented type of
Forman-Ricci curvature, in directed complex networks.
