Discrete Ricci curvatures for directed networks. (January 2019)

• Record Type: Journal Article
• Authors: Saucan, Emil; Sreejith, R.P.; Vivek-Ananth, R.P.; Jost, Jürgen; Samal, Areejit
• Is Part Of: Chaos, solitons and fractals. Volume 118(2019)
• Page Start: 347
• Page End: 360
• Publication Date: 2019-01
• Subjects: Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2018.11.031
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: Extension of the two discrete notions of Ricci curvature, Forman–Ricci and Ollivier–Ricci, to directed networks. Adaptation of the Forman–Ricci curvature to define an augmented version which also accounts for two-dimensional faces capturing higher-order correlations between vertices in directed networks. Evaluation of the three discrete notions of Ricci curvature, namely, Forman–Ricci, Augmented Forman–Ricci and Ollivier–Ricci curvature for directed edges in model and real-world 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.