Forman–ricci Curvature for Hypergraphs

Abstract

Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman–Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman–Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman–Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman–Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman–Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.

Chemical space: limits, evolution and modelling of an object bigger than our universal library

“…22 Although these models have been studied for hypergraphs, [106][107][108][109][110][111][112][113][114][115][116] there are only few accounts of dynamical models for directed hypergraphs. [117][118][119] At any rate, a general setting for dynamical directed hypergraphs requires probabilistic rules to include new substances (vertices) in the network and also to wire substances by chemical reactions, that is rules to include new hyperedges in the hypergraph. These rules may come from two different sources.…”

Spaces of mathematical chemistry

“…The hypergraph description of chemical reactions opens a new field of research in mathematical chemistry, as the mathematics of these structures is still to be developed. As discussed in [35,37], only a few mathematical properties of these structures have been studied, for example vertex and hyperedge degrees [35], clustering coefficients [38,39], spectral properties [40], curvatures [37] and more recently the Erdős-Rényi model for the random hypergraph [35]. Nevertheless, other aspects, including different random models, measures of assortativity, and betweenness centrality, among others, remain unexplored, as well as further curvatures and network geometry notions as those pioneered by Jost and collaborators [30,37,[41][42][43][44].…”

“…As discussed in [35,37], only a few mathematical properties of these structures have been studied, for example vertex and hyperedge degrees [35], clustering coefficients [38,39], spectral properties [40], curvatures [37] and more recently the Erdős-Rényi model for the random hypergraph [35]. Nevertheless, other aspects, including different random models, measures of assortativity, and betweenness centrality, among others, remain unexplored, as well as further curvatures and network geometry notions as those pioneered by Jost and collaborators [30,37,[41][42][43][44]. On top of this mathematics to develop, the connection with chemistry is central, that is the interpretation and implications of those mathematical properties for the study of the chemical space.…”

The Twitter Explorer: a Framework for Observing Twitter through Interactive Networks

“…In order to disseminate the framework and attract new audiences to the field of data-driven research, vignettes (use-cases) will be designed to showcase the twitter explorer's use in social science research. Furthermore, it is planned to add the possibility of exploring recently developed measures such as graph curvatures which can provide new insights to the analysis of social networks [27].…”