Uplifting edges in higher-order networks: Spectral centralities for non-uniform hypergraphs

Spectral analysis of networks states that many structural properties of graphs, such as the centrality of their nodes, are given in terms of their adjacency matrices. The natural extension of such spectral analysis to higher-order networks is strongly limited by the fact that a given hypergraph could have several different adjacency hypermatrices, and hence the results obtained so far are mainly restricted to the class of uniform hypergraphs, which leaves many real systems unattended. A new method for analyzing non-linear eigenvector-like centrality measures of non-uniform hypergraphs was presented in this paper that could be useful for studying properties of $\mathcal{H}$-eigenvectors and $\mathcal{Z}$-eigenvectors in the non-uniform case. In order to do so, a new operation——the uplift——was introduced, incorporating auxiliary nodes in the hypergraph to allow for a uniform-like analysis. We later argued why this was a mathematically sound operation, and we furthermore used it to classify a whole family of hypergraphs with unique Perron-like $\mathcal{Z}$-eigenvectors. We supplemented the theoretical analysis with several examples and numerical simulations on synthetic and real datasets: On the latter, we find a clear improvement over the existing methods, specially in cases where there is a huge disparity between the structure at each order, and on the former, we find that regardless of the chosen uniformization scheme, the nodes were similarly ranked.