Title: Centralities in Simplicial Complexes,
Abstract: Complex networks can be used to represent complex systems which originate in
the real world. Here we study a transformation of these complex networks into
simplicial complexes, where cliques represent the simplices of the complex. We
extend the concept of node centrality to that of simplicial centrality and
study several mathematical properties of degree, closeness, betweenness,
eigenvector, Katz, and subgraph centrality for simplicial complexes. We study
the degree distributions of these centralities at the different levels. We also
compare and describe the differences between the centralities at the different
levels. Using these centralities we study a method for detecting essential
proteins in PPI networks of cells and explain the varying abilities of the
centrality measures at the different levels in identifying these essential
proteins. The paper is written in a self-contained way, such that it can be
used by practitioners of network theory as a basis for further developments.