Three hypergraph eigenvector centralities
This work addresses the problem of ranking entities in complex systems with multi-way interactions for researchers in network analysis, though it is incremental as it builds on existing tensor theory.
The authors extended eigenvector centrality from graphs to uniform hypergraphs to better model multi-way interactions, developing three tensor eigenvector centralities that reveal different information in real-world datasets like n-gram frequencies and drug combinations.
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.