  We adapt Forman's discretization of Ricci curvature to the case of undirected
networks, both weighted and unweighted, and investigate the measure in a
variety of model and real-world networks. We find that most nodes and edges in
model and real networks have a negative curvature. Furthermore, the
distribution of Forman curvature of nodes and edges is narrow in random and
small-world networks, while the distribution is broad in scale-free and
real-world networks. In most networks, Forman curvature is found to display
significant negative correlation with degree and centrality measures. However,
Forman curvature is uncorrelated with clustering coefficient in most networks.
Importantly, we find that both model and real networks are vulnerable to
targeted deletion of nodes with highly negative Forman curvature. Our results
suggest that Forman curvature can be employed to gain novel insights on the
organization of complex networks.
