Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.

中文翻译：

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单纯复形中的中心性测度：拓扑数据分析在网络科学中的应用

社会科学、生物和生物医学科学或计算机科学中的许多真实网络都具有反映多体相互作用的简单复合物的固有结构。因此，为了分析单纯复网络的拓扑和动力学特性，需要提出针对单纯的中心性度量。许多经典的复杂网络中心性基于节点的度数，因此为了定义单纯形的度中心性度量（这将表征单纯形网络中单纯形社区的相关性），单纯形之间邻接的不同定义是必需的。这些笔记的目的有三个：首先，我们将使用最近引入的高阶单纯度的概念来提出单纯复形中基于度的新中心性度量。这些理论中心性度量，例如单纯度中心性或特征向量中心性，不仅可以研究单纯社区的相关性及其在单纯网络中的高阶连接的质量，而且可能有助于阐明拓扑和单纯网络的动力学特性；其次，我们定义了单纯复形中的步行和距离的概念，以研究单纯网络的连通性，并将众所周知的接近性和介数中心性概括为单纯性情况（例如需要研究单纯性社区在术语方面的相关性）其传递信息的能力）；第三，我们为单纯网络中的单纯提出了一个新的聚类系数，与目前已知的不同，它概括了顶点的标准图聚类。该度量对于了解简单网络在其单纯社区方面的密度是必不可少的。