  The characterization of complex networks with tools originating in geometry,
for instance through the statistics of so-called Ricci curvatures, is a well
established tool of network science. There exist various types of such Ricci
curvatures, capturing different aspects of network geometry. In the present
work, we investigate Bakry-\'Emery-Ricci curvature, a notion of discrete Ricci
curvature that has been studied much in geometry, but so far has not been
applied to networks. We explore on standard classes of artificial networks as
well as on selected empirical ones to what the statistics of that curvature are
similar to or different from that of other curvatures, how it is correlated to
other important network measures, and what it tells us about the underlying
network. We observe that most vertices typically have negative curvature.
Random and small-world networks exhibit a narrow curvature distribution whereas
other classes and most of the real-world networks possess a wide curvature
distribution. When we compare Bakry-\'Emery-Ricci curvature with two other
discrete notions of Ricci-curvature, Forman-Ricci and Ollivier-Ricci curvature
for both model and real-world networks, we observe a high positive correlation
between Bakry-\'Emery-Ricci and both Forman-Ricci and Ollivier-Ricci curvature,
and in particular with the augmented version of Forman-Ricci curvature.
Bakry-\'Emery-Ricci curvature also exhibits a high negative correlation with
the vertex centrality measure and degree for most of the model and real-world
networks. However, it does not correlate with the clustering coefficient. Also,
we investigate the importance of vertices with highly negative curvature values
to maintain communication in the network. The computational time for
Bakry-\'Emery-Ricci curvature is shorter than that required for Ollivier-Ricci
curvature but higher than for Augmented Forman-Ricci curvature.
