Modern quantitative challenges require to tackle problems on increasingly complex systems in which the relationships between the comprised entities cannot be modelled in a simple pairwise fashion, as graphs do. Such approximation of higher-order relations may lead to a substantial loss of information, hence the need to use more general models than graphs. The most natural choice is to use hypergraphs, discrete structures able to capture $k$-adic relationships among the entities participating in the problem, modelled as vertices, by grouping them in non-empty sets which constitute the hyperedges of the hypergraph. Since one of the most desirable abilities in this context is to quantify the difference between two such high-order systems, devising distance metrics between hypergraphs becomes of the utmost importance. In this paper, we aim at tackling precisely this problem. Motivated by our previous work on graphs, we propose two distance measures between attributed hypergraphs and we prove that they satisfy the properties of a metric. Both metrics are based on the notion of the maximal common subhypergraph.

现代定量挑战需要解决日益复杂的系统上的问题，在这些系统中，所包含的实体之间的关系不能像图那样以简单的成对方式建模。这种高阶关系的近似可能会导致大量的信息丢失，因此需要使用比图更通用的模型。最自然的选择是使用超图，离散结构能够捕获$克$-参与问题的实体之间的adic关系，建模为顶点，通过将它们分组在构成超图的超边的非空集合中。由于在这种情况下最理想的能力之一是量化两个此类高阶系统之间的差异，因此设计超图之间的距离度量变得至关重要。在本文中，我们旨在精确解决这个问题。受我们之前关于图的工作的启发，我们提出了两个属性超图之间的距离度量，并证明它们满足度量的属性。这两个指标都基于最大公共子超图的概念。