Matrix centrality for annotated hypergraphs

The identification of central nodes within networks constitutes a task of fundamental importance in various disciplines, and it is an extensively explored problem within the scientific community. Several scalar metrics have been proposed for classic networks with dyadic connections, and many of them have later been extended to networks with higher-order interactions. We here introduce two novel measures for annotated hypergraphs: that of matrix centrality and that of role centrality. These concepts are formulated for hypergraphs where the roles of nodes within hyper-edges are explicitly delineated. Matrix centrality entails the assignment of a matrix to each node, whose dimensions are determined by the size of the largest hyper-edge in the hypergraph and the number of roles defined by the annotated hypergraph's labeling function. This formulation facilitates the simultaneous ranking of nodes based on both hyper-edge size and role type. The second concept, role centrality, involves assigning a vector to each node, the dimension of which equals the number of roles specified. This metric enables the identification of pivotal nodes across different roles without distinguishing hyper-edge sizes. Through the application of these novel centrality measures to a range of synthetic and real-world examples, we demonstrate their efficacy in providing enhanced insights into the structural characteristics of the systems under consideration.