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, the two discretizations of the Ricci curvature, Forman-Ricci and Ollivier-Ricci, 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 wide-range of both model and real-world networks shows that the two discretizations of Ricci curvature are statistically well correlated. 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 networks whenever coarse analysis suffices.
The relationship between the two approaches have been explained in REF .
Comparative analysis of two discretizations of Ricci curvature for complex networks
Scientific Reports 8(1): 8650 (2018)
Computer Science Mathematics Medicine