Forman-Ricci Curvature for Hypergraphs

In contrast to graph-based models for complex networks, hypergraphs are more general structures going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their structures have been devised and there is undergoing research for devising them for hypergraphs. Forman-Ricci curvature is a statistics for graphs, which is based on Riemannian geometry, and that stresses the relational character of vertices in a network through the analysis of edges rather than vertices. In spite of the different applications of this curvature, it has not yet been formulated for hypergraphs. Here we devise the Forman-Ricci curvature for directed and undirected hypergraphs, where the curvature for graphs is a particular case. We report its upper and lower bounds and the respective bounds for the graph case. The curvature quantifies the trade-off between hyperedge(arc) size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calculated the curvature for two large networks: Wikipedia vote network and \emph{Escherichia coli} metabolic network. In the first case the curvature is ruled by hyperedge size, while in the second by 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 phosphorilation is the metabolic bottle neck reaction but that the reverse reaction is not that central for the metabolism.