Comparing the principal eigenvector of a hypergraph and its shadows

Abstract

Graphs (i.e., networks) have become an integral tool for the representation and analysis of relational data. Advances in data gathering have led to multi-relational data sets which exhibit greater depth and scope. In certain cases, this data can be modeled using a hypergraph. However, in practice analysts typically reduce the dimensionality of the data (whether consciously or otherwise) to accommodate a traditional graph model. In recent years spectral hypergraph theory has emerged to study the eigenpairs of the adjacency hypermatrix of a uniform hypergraph. We show how analyzing multi-relational data, via a hypermatrix associated to the aforementioned hypergraph, can lead to conclusions different from those when the data is projected down to its co-occurrence matrix. To this end we consider how the principal eigenvector of a hypergraph and its shadow can vary in terms of their spectral rankings, Pearson/Spearman correlation coefficient, and Chebyshev distance. In particular, we provide an example of a uniform hypergraph where the most central vertex (à la eigencentrality) changes depending on the order of the associated matrix. To the best of our knowledge this is the first known hypergraph to exhibit this property. We further show that the aforementioned eigenvectors have a high Pearson correlation but are uncorrelated under the Spearman correlation coefficient.