  Eigenvector centrality is a standard network analysis tool for determining
the importance of (or ranking of) entities in a connected system that is
represented by a graph. However, many complex systems and datasets have natural
multi-way interactions that are more faithfully modeled by a hypergraph. Here
we extend the notion of graph eigenvector centrality to uniform hypergraphs.
Traditional graph eigenvector centralities are given by a positive eigenvector
of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius
theorem under some mild conditions. The natural representation of a hypergraph
is a hypermatrix (colloquially, a tensor). Using recently established
Perron-Frobenius theory for tensors, we develop three tensor eigenvectors
centralities for hypergraphs, each with different interpretations. We show that
these centralities can reveal different information on real-world data by
analyzing hypergraphs constructed from n-gram frequencies, co-tagging on stack
exchange, and drug combinations observed in patient emergency room visits.

Social and Information Networks;Machine Learning;Physics and Society