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

Gonzalo Contreras-Aso, Cristian Pérez-Corral, Miguel Romance

TL;DR

This work tackles the challenge of defining spectral centralities for non-uniform hypergraphs by introducing an uplift operation that adds auxiliary nodes to raise hyperedges to a uniform order, enabling analysis with $\mathcal{H}$-eigenvector centrality. It develops UPHEC, a unified centrality measure obtained by uplifting lower-order edges and projecting higher-order ones, and proves existence and uniqueness results under strong connectivity. The authors also establish a Perron-like $\mathcal{Z}$-eigenvector framework for uplifted graphs, provide sufficient conditions for uniqueness, and validate the approach with extensive real and synthetic data, showing consistent and scalable centrality rankings. The methodology bridges non-uniform higher-order networks with established tensor-based centralities, offering a practical toolkit for identifying influential nodes across complex systems while maintaining computational efficiency and interpretability.

Abstract

Spectral analysis of networks states that many structural properties of graphs, such as 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, 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 analysing non-linear eigenvector-like centrality measures of non-uniform hypergraphs is 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 $\textit{uplift}$ - is introduced, incorporating auxiliary nodes in the hypergraph to allow for a uniform-like analysis. We later argue why this is a mathematically sound operation, and we furthermore use it to classify a whole family of hypergraphs with unique Perron-like $\mathcal{Z}$-eigenvectors. We supplement the theoretical analysis with several examples and numerical simulations on synthetic and real datasets.

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

TL;DR

This work tackles the challenge of defining spectral centralities for non-uniform hypergraphs by introducing an uplift operation that adds auxiliary nodes to raise hyperedges to a uniform order, enabling analysis with -eigenvector centrality. It develops UPHEC, a unified centrality measure obtained by uplifting lower-order edges and projecting higher-order ones, and proves existence and uniqueness results under strong connectivity. The authors also establish a Perron-like -eigenvector framework for uplifted graphs, provide sufficient conditions for uniqueness, and validate the approach with extensive real and synthetic data, showing consistent and scalable centrality rankings. The methodology bridges non-uniform higher-order networks with established tensor-based centralities, offering a practical toolkit for identifying influential nodes across complex systems while maintaining computational efficiency and interpretability.

Abstract

Spectral analysis of networks states that many structural properties of graphs, such as 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, 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 analysing non-linear eigenvector-like centrality measures of non-uniform hypergraphs is presented in this paper that could be useful for studying properties of -eigenvectors and -eigenvectors in the non-uniform case. In order to do so, a new operation - the - is introduced, incorporating auxiliary nodes in the hypergraph to allow for a uniform-like analysis. We later argue why this is a mathematically sound operation, and we furthermore use it to classify a whole family of hypergraphs with unique Perron-like -eigenvectors. We supplement the theoretical analysis with several examples and numerical simulations on synthetic and real datasets.
Paper Structure (22 sections, 7 theorems, 30 equations, 8 figures, 4 tables)

This paper contains 22 sections, 7 theorems, 30 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The eigenvector centrality in graphs:
  4. Notions from algebraic hypergraph theory:
  5. Hypergraph spectral centralities: State of the Art
  6. Vectorial characterizations
  7. Embracing non-uniformity
  8. The uplift
  9. An uplifted nuance.
  10. The ZEC problem.
  11. Uplift + $\mathcal{H}$-eigenvectors: spectral centrality in non-uniform hypergraphs
  12. The pairwise case
  13. Uplifting and projecting hypergraphs
  14. Joining everything together:
  15. Notes on computational complexity:
  16. ...and 7 more sections

Key Result

Lemma 3.4

Let $H=(V,E)$ be a strongly connected hypergraph whose maximum hyperedge size is $M$ and let $m>M$. Then, the uplifted hypergraph $\widetilde{H}$ is strongly connected.

Figures (8)

  • Figure 1: Uplift of the hypergraph $H=(\{1,2,3,4,5\},\{\{1,2,3\},\{2,4\},\{3,5\}\})$ to $\widetilde{H}$.
  • Figure 2: Panel a) shows the pairwise graph $G=(V,E)$ with 7 nodes whose Eigenvector Centrality (EC) and the HEC of its uplift to order 3 (HEC) are calculated. Panel b) shows the centrality values associated to the $\ell_1$-normalized centralities in a parallel coordinate plot. The effect of the homogenization can be clearly seen, as well as the crossing in ranking between nodes $1$ and $2$.
  • Figure 3: Example hypergraph and its three possible uniformizations at $p=2,3,4$.
  • Figure 4: Kendall's tau correlation coefficient between the whole rankings obtained in each of the method, for the tags_ask_ubuntu (left) and hypertext-conference (right) datasets. Methods are labelled as U2, U3, U4, U5 for the UPHEC case; H2, H3, H4, H5 for the HEC at each order, and B3, B4, B5 for the blowup uniformization discussed above. U2 and B2 are equal as they only include the projection, and are therefore shown together.
  • Figure 5: Kendall's tau correlation coefficient between the whole rankings obtained by the UPHEC and blowup uniformizations (with the same labelling conventions as in Figure \ref{['fig:KT-double_heatmap']}) for the contact-primary-school (top-left), contact-high-school (top-right), sfhh-conference (bottom-left) and diseasome (bottom-right) datasets.
  • ...and 3 more figures

Theorems & Definitions (23)

  • Definition 2.1: Irreducible tensor qi2017tensor
  • Definition 2.2: Weakly irreducible tensor qi2017tensor
  • Definition 3.1: Uplifted hypergraph at order $m$
  • Remark 3.2
  • Definition 3.3: Uplifted tensor of a hypergraph
  • Lemma 3.4: Strong connectedness of the uplifted hypergraph
  • Definition 4.1: $m$-UHEC
  • Theorem 4.2: Existence and uniqueness of the UHEC
  • proof
  • Definition 4.3: Projected hypergraph
  • ...and 13 more