  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.
