Node and edge nonlinear eigenvector centrality for hypergraphs

Evaluating the importance of nodes and hyperedges in hypergraphs is relevant to link detection, link prediction and matrix completion. Here, the authors define a family of nonlinear eigenvector centrality measures for both edges and nodes in hypergraphs, propose an algorithm to calculate them, and illustrate their application on real-world data sets. Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.