Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.

翻译：图中顶点间的相似性概念，如结构等价与正则等价，是复杂网络科学中聚类工具的主要组成部分之一。我们将结构等价与正则等价推广至无向超图，并通过邻域图与Ollivier-Ricci曲率刻画无向图及超图中的结构等价与正则等价。这一刻画为这些相似性概念提供了新的理解视角，开辟了探索它们的新途径。这些刻画还使得构建可能广泛的规则划分族成为可能，从而为这一迄今计算上极具挑战性的任务提供了新的解决路径。