Discrete Ricci curvatures for directed 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.

arXiv comment26 pages, 5 figures, 4 tables, supplementary information available upon request from authors. The present manuscript incorporates some unpublished results from our unpublished preprint arXiv:1605.04662. In addition, the manuscript extends the notions of augmented Forman-Ricci curvature and Ollivier-Ricci curvature to directed graphs

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• Network-centric indicators for fragility in global financial indices
• Hochschild homology, and a persistent approach via connectivity digraphs
• Forman-Ricci curvature and Persistent homology of unweighted complex networks
• Complex Networks in the Framework of Nonassociative Geometry
• A Cosine Rule-Based Discrete Sectional Curvature for Graphs
• Hypernetworks: From Posets to Geometry