Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. The relevance of an agent in a graph network is given in terms of its degree, and in a simplicial network there are already notions of adjacency and degree for simplices that, as far as we know, are not valid for comparing simplices in different dimensions. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined. As for the graph or combinatorial Laplacian, the multi-combinatorial Laplacian is shown to be an effective tool for calculating the higher-order degrees presented here. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.

网络科学提供了一种通用的形式主义，用于基于代理之间的成对交互来建模和研究复杂的系统。但是，社会，生物学或计算机科学中的许多实际网络都涉及两个以上主体之间的相互作用，因此具有简单复合体的固有结构。图网络中的主体的相关性是根据其程度给出的，在一个简单网络中，已经存在邻接词的邻接度和度的概念，据我们所知，这些概念对于比较不同维度的邻接词是无效的。我们为简单复合体中的单纯形提出了更高阶邻接度的新概念，从而允许它们和它们的面孔之间进行任何尺寸比较。我们在定向简单复数中引入了多参数边界和共边界算子，并定义了一种新颖的多组合拉普拉斯算子。至于图或组合拉普拉斯算子，多组合拉普拉斯算子是显示此处计算的高阶度的有效工具。为了说明这些理论结果的潜在应用，我们通过研究来自不同领域（例如共同作者网络，共同发起人）的17个真实数据集中的与这些简单程度相关的分布，对简单复杂网络中的高阶连通性进行了结构分析。国会法案，学校联系人，药物滥用警告网络，电子邮件网络或出版物以及在线论坛的用户。我们发现了丰富多样的高阶连接结构，并观察到相同类型的数据集反映了相似的高阶协作模式。此外，我们表明，如果我们使用所谓的最大简单度（它计算包含单纯形及其所有严格子社区的不同最大社区），那么其程度分布通常与经典节点度分布。