Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we show that Ricci curvature concepts, inspired by the corresponding notions of Forman and Ollivier for graphs, are powerful tools for probing the local geometry of hypergraphs. In fact, these two curvature concepts complement each other in the identification of specific connectivity motifs. In order to have a baseline model with which we can compare empirical data, we introduce a random model to generate directed hypergraphs and study properties such as degree of nodes and edge curvature, using numerical simulations. We can then see how our notions of curvature can be used to identify connectivity patterns in the metabolic network of E. coli that clearly deviate from those of our random model. Specifically, by applying hypergraph shuffling to this metabolic network we show that the changes in the wiring of a hypergraph can be detected by Forman Ricci and Ollivier Ricci curvatures.

中文翻译：

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随机和经验有向超网络的Ricci曲率

实际系统中的关系通常不是二进制的，而是更高阶的，因此不能用图如实地建模，而是需要超图。在这项工作中，我们系统地开发了用于分析超图的几何和动力学的形式化工具。特别是，我们表明，受Forman和Ollivier对应图的概念启发的Ricci曲率概念是探测超图的局部几何的强大工具。实际上，这两个曲率概念在特定连通性图案的识别中彼此互补。为了拥有一个可以与经验数据进行比较的基线模型，我们引入了一个随机模型来生成有向超图，并使用数值模拟研究诸如节点度和边缘曲率之类的属性。明显不同于我们随机模型的大肠杆菌。具体而言，通过将超图混排应用于此代谢网络，我们表明，可以通过Forman Ricci和Ollivier Ricci曲率检测到超图布线的变化。