Bakry–Émery–Ricci curvature: an alternative network geometry measure in the expanding toolbox of graph Ricci curvatures.

This article focuses on the investigation of Bakry–Émery–Ricci curvature, a discrete notion of Ricci curvature defined on vertices, and its application to complex networks. The study compares Bakry–Émery–Ricci curvature with other discrete Ricci curvatures—Forman–Ricci, Augmented Forman–Ricci, and Ollivier–Ricci curvatures—across various model and real-world undirected, unweighted networks, analyzing their distributions, correlations with network measures (degree, clustering coefficient, centralities), and implications for network robustness. Results indicate that Bakry–Émery–Ricci curvature is generally negatively valued, highly correlated with Augmented Forman–Ricci and Ollivier–Ricci curvatures, and computationally less demanding than Ollivier–Ricci curvature but more so than Augmented Forman–Ricci curvature. The curvature effectively identifies vertices critical for maintaining network connectivity, with scalar curvature (sum of incident edge curvatures) showing even stronger predictive power for network robustness. The authors suggest Augmented Forman–Ricci curvature as a practical alternative for empirical network analysis, especially in high edge density networks.