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

Understand

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.