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Many real networks in social sciences, biological and biomedical sciences orcomputer science have an inherent structure of simplicial complexes reflectingmany-body interactions. Therefore, to analyse topological and dynamicalproperties of simplicial complex networks centrality measures for simplicesneed to be proposed. Many of the classical complex networks centralities arebased on the degree of a node, so in order to define degree centrality measuresfor simplices (which would characterise the relevance of a simplicial communityin a simplicial network), a different definition of adjacency between simplicesis required. The aim of these notes is threefold: first we will use therecently introduced notions of higher order simplicial degrees to propose newdegree based centrality measures in simplicial complexes. These theoreticalcentrality measures, such as the simplicial degree centrality or theeigenvector centrality would allow not only to study the relevance of asimplicial community and the quality of its higher-order connections in asimplicial network, but also they might help to elucidate topological anddynamical properties of simplicial networks; sencond, we define notions ofwalks and distances in simplicial complexes in order to study connectivity ofsimplicial networks and to generalise, to the simplicial case, the well knowncloseness and betweenness centralities (needed for instance to study therelevance of a simplicial community in terms of its ability of transmittinginformation); third, we propose a new clustering coefficient for simplices in asimplicial network, different from the one knows so far and which generalisesthe standard graph clustering of a vertex. This measure should be essential toknow the density of a simplicial network in terms of its simplicialcommunities.

Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science