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To keep track of the importance of each vertex and each hyperedge of ahypergraph, we introduce some positive valued functions on the set of verticesand hyperedges, respectively. The terms vertex centrality and hyperedgecentrality, respectively, refer to these functions on the set of vertices andhyperedges. The concepts of centrality vary according to the sense ofimportance. We introduce and investigate two eigenvector-based vertexcentralities and one eigenvector-based hyperedge centrality here. We introducesome distance-based hypergraph centrality on the set of vertices by employing apseudo metric called unit-distance. We also explore random walk-basedhypergraph centralities. Our study shows that these centrality functions becomeconstant on some clusters of vertices and thus provide some clusters of equallyimportant vertices. Some of these clusters appear only in networks withmulti-body interactions and disappear in networks with only pair-wiseinteractions.

On hypergraph centrality: Some clusters of vertices with equal importance