Comparative analysis of two discretizations of Ricci curvature for complex networks

TL;DR

This paper empirically compares two discrete Ricci curvature notions, Forman-Ricci and Ollivier-Ricci, revealing high correlation and practical implications for network analysis and computational efficiency.

Contribution

It provides the first extensive empirical comparison of Forman-Ricci and Ollivier-Ricci curvatures across diverse networks, highlighting their correlation and practical interchangeability.

Findings

High correlation between the two curvature measures in many networks.

Augmented Forman-Ricci curvature increases correlation, especially in real networks.

Forman-Ricci curvature can replace Ollivier-Ricci for faster analysis in large networks.

Abstract

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.