Discrete Ricci curvatures for directed networks

TL;DR

This paper extends Forman's Ricci curvature to directed networks, providing a mathematically elegant measure that incorporates weights and directions, and compares it with Ollivier-Ricci curvature, also exploring higher-order correlations.

Contribution

The paper introduces a new formulation of Forman-Ricci curvature for directed networks, including vertex and edge weights, and compares it with Ollivier-Ricci curvature, also considering higher-order structures.

Findings

Forman-Ricci curvature for directed networks incorporates weights and directions.

Comparison shows differences between Forman-Ricci and Ollivier-Ricci curvatures.

Augmented curvature accounts for higher-order correlations in directed complex networks.

Abstract

A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman-Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman-Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman-Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier-Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman's original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman-Ricci curvature, in directed complex networks.