On λ-Forman-Ricci curvature of networks

Several discrete versions of Ricci curvature have been proposed since the geometrical properties of a network are used to understand important information associated with it. In this paper we obtain the main properties of the λ-Forman-Ricci curvature, a concept that generalizes and integrates the Forman-Ricci curvature and the augmented Forman-Ricci curvature. We show that this definition captures the essence of Ricci curvature in Riemannian manifolds, by proving discrete analogues of important results in geometry. Also, we study the integral λ-Forman-Ricci curvature, obtaining a kind of Gauss-Bonnet formula, and we study this integral curvature in the context of random networks.