  We have performed an empirical comparison of two distinct notions of discrete
Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and
Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci
curvature were developed based on different properties of the classical smooth
notion, and thus, the two notions shed light on different aspects of network
structure and behavior. Nevertheless, our extensive computational analysis in a
wide range of both model and real-world networks shows that the two
discretizations of Ricci curvature are highly correlated in many networks.
Moreover, we show that if one considers the augmented Forman-Ricci curvature
which also accounts for the two-dimensional simplicial complexes arising in
graphs, the observed correlation between the two discretizations is even
higher, especially, in real networks. Besides the potential theoretical
implications of these observations, the close relationship between the two
discretizations has practical implications whereby Forman-Ricci curvature can
be employed in place of Ollivier-Ricci curvature for faster computation in
larger real-world networks whenever coarse analysis suffices.
