Forman curvature for complex networks

Abstract

A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced the Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of complex undirected networks. However, many important realworld networks are inherently directed in nature, and the definition of the Forman curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman curvature of a directed edge elegantly incorporates node weights, edge weights and edge direction. By applying the Forman curvature for directed networks to a variety of model and real-world directed networks, we show that the measure can be used to characterize the structure of complex directed networks. Furthermore, our results also hold in real directed networks which are weighted or spatial in nature. These results in combination with our previous results suggest that the Forman curvature can be readily employed to study the organization of both directed and undirected complex networks. * jost@mis.mpg.de † this also makes the measure suitable for analysis of both unweighted and weighted networks [21]. Since Forman curvature represents a discretization of the classical Ricci curvature which is intrinsically associated with edges of a network, this notion of curvature does not necessitate the technical artifice of extending a measure for the curvature of nodes to the edges [21]. Thus, Forman curvature can be exploited for edge-based analysis of complex networks. Given the definition of the Forman curvature of an edge, one can easily define the Forman curvature of a node in the network by summing or averaging the curvatures of its adjacent edges, somewhat analogous to the concept of scalar curvature in Riemannian geometry [23]. We remark that the Forman curvature for an edge is a local measure dependent on weights of adjacent nodes and edges in the network [21]. Still Forman curvature, a local geometric characteristic, can provide deep insights on the global topology of the network [30]. Moreover, two networks with the same degree distribution can have very different distributions of Forman curvature (Fig. 1).Although, we have successfully introduced Forman curvature to undirected networks [21], several important real networks in nature and society are inherently directed in nature. These include the metabolic networks [7,33,34], gene regulatory networks [35], signaling networks [36], neural networks [37], the world wide web (WWW) [38], online social networks [10] and transportation networks [39,40]. However, the two different discretizations of the Ricci curvature, Ollivier-Ricci curvature and Forman-Ricci curvature, have been...

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