Eigenvector centrality in simplicial complexes of hypergraphs

Hypergraph is the model of relations lying in clusters of objects. Identifying vital nodes is a fundamental problem in the analysis of the hypergraph. To reflect the multilayer feature of the hypergraph, in this paper, we deconstruct the hypergraph into a simplicial complex and analyze the homological dual relations of boundary and coboundary between simplices. For clarity, these two relations are summarized into a bidirectional graph, called the simplicial diagram, which provides a global framework for the exploration of the hypergraph. To determine the node importance in the hypergraph, we propose a parameter-free eigenvector centrality for weighted hypergraphs in terms of a simplicial complex, named Simplicial DualRank centrality. For each simplex, we define two indices of importance, the inner centrality and the outer centrality. Inner centrality transmits according to the relation of coboundary, which converts to outer centrality at the hyperlinks; in duality, outer centrality transmits according to the relation of boundary, which converts to inner centrality at the nodes. Therefore, a circuit of centrality is constructed on the simplicial diagram, the steady state of which defines the Simplicial DualRank centrality of all the simplices in the hypergraph. Moreover, we apply the Simplicial DualRank centrality to weighted complex networks, which results in a variant of the classical eigenvector centrality. Finally, experimental results in a science collaboration dataset show that the Simplicial DualRank can identify Nobel laureates from the prize-winning papers in Physics, top scientists should select collaborators more carefully to maintain their research quality, and scholars tend to find relatively effective collaborations in their future research.