Networks are important structures used to model complex systems where interactions take place. In a basic network model, entities are represented as nodes, and interaction and relations among them are represented as edges. However, in a complex system, we cannot describe all relations as pairwise interactions, rather should describe as higher-order interactions. Hypergraphs are successfully used to model higher-order interactions in complex systems. In this paper, we present two new hypergraph Laplacians based on diffusion framework. Our Laplacians take the relations between higher-order interactions into consideration, hence can be used to model diffusion on hypergraphs not only between vertices but also higher-order structures. These Laplacians can be employed in different network mining problems on hypergraphs, such as social contagion models on hypergraphs, influence study on hypergraphs, and hypergraph classification, to list a few.

中文翻译：

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扩散框架中的超图拉普拉斯算子

网络是用于对发生交互的复杂系统进行建模的重要结构。在基本网络模型中，实体表示为节点，而它们之间的交互和关系表示为边缘。但是，在复杂的系统中，我们不能将所有关系描述为成对相互作用，而应该描述为高阶相互作用。超图已成功用于建模复杂系统中的高阶交互。在本文中，我们提出了两个基于扩散框架的超图Laplacian。我们的拉普拉斯算子考虑了高阶相互作用之间的关系，因此不仅可以用于顶点之间的超图扩散，还可以用于建模高阶结构上的扩散。这些拉普拉斯算子可以用于超图上的不同网络挖掘问题，