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.

中文翻译：

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FORMAN-RICCI 曲率的超图

超图用作复杂网络的模型，它捕获比二元关系更一般的结构。对于图表，已经设计了广泛的统计数据来衡量其结构的不同方面。超图在这方面落后。Forman-Ricci 曲率是基于黎曼几何的图的统计量，它通过关注边而不是顶点来强调网络中顶点的关系特征。尽管该度量已成功应用于图，但尚未为超图引入 Forman-Ricci 曲率。在这里，我们定义了有向和无向超图的 Forman-Ricci 曲率，以便将图的曲率恢复为一种特殊情况。它量化了超边（弧）大小和超边（弧）顶点在其他超边（弧）中的参与程度之间的权衡。在这里，我们确定了一般超图和特别是图的 Forman-Ricci 曲率的上限和下限。然后将该度量应用于两个大型网络：维基百科投票网络和细菌的代谢网络大肠杆菌. 在第一种情况下，曲率由超边的大小决定，而在第二种情况下，曲率由超边度决定。我们发现参与维基百科选举的用户数量与经验丰富的用户的参与密切相关。代谢网络的曲率值允许检测冗余和瓶颈反应。发现 ADP 磷酸化是代谢的瓶颈反应，但逆反应对于代谢并不同样重要。此外，我们展示了 Forman-Ricci 曲率在超图中量化分类性的效用，并通过研究三个代谢网络来说明这一想法。