A Survey on Centrality and Importance Measures in Hypergraphs: Categorization and Empirical Insights

TL;DR

The paper addresses the challenge of identifying central entities in hypergraphs by proposing a three-way taxonomy—structural, functional, and contextual—for 39 centrality and importance measures. It grounds the taxonomy in preliminaries, surveys, and categorization across subtypes, and then provides an empirical study of measure similarity and computation time on ten real-world hypergraphs. Key findings reveal strong global correlations among measures but divergent top-ranked selections, with distinct clusters and a few singleton measures like eigenvector centrality and hypercoreness. The work offers practical guidance for selecting measures based on accuracy-efficiency tradeoffs and outlines future directions toward axiomatic foundations, unified path frameworks, non-uniform spectral operators, generalized hypergraphs, and benchmarking standards.

Abstract

Identifying central entities and interactions is a fundamental problem in network science. While well-studied for graphs (pairwise relations), many biological and social systems exhibit higher-order interactions best modeled by hypergraphs. This has led to a proliferation of specialized hypergraph centrality measures, but the field remains fragmented and lacks a unifying framework. This paper addresses this gap by providing the first systematic survey of 39 distinct measures. We introduce a novel taxonomy classifying them as: (1) structural (topology-based), (2) functional (impact on system dynamics), or (3) contextual (incorporating external features). We also present an experimental assessment comparing their empirical similarity and computation time. Finally, we discuss applications, establishing a coherent roadmap for future research in this area.

Figures (5)

• Figure 1: (a) Example co-authorship hypergraph and (b) corresponding node centrality and importance values. Different measures emphasize different structural roles, resulting in distinct rankings. For example, betweenness centrality (\ref{['cent:betweenness']}) highlights author L due to its bridging position across multiple publication groups, whereas hypercoreness (\ref{['cent:core']}) assigns the highest scores to F, P, and S, reflecting the tightly connected triadic region they form.
• Figure 2: Taxonomy of hypergraph centrality and importance measures. The framework distinguishes three major categories: structural measures, which rely on the combinatorial structure of the hypergraph; functional measures, which evaluate importance in terms of system behavior; and contextual measures, which integrate information beyond structure. Each category is further divided into sub-categories that capture specific methodological choices.
• Figure 3: Correlation between node centrality and importance measures in terms of (a) Pearson correlation coefficient, (b) Spearman rank correlation coefficient, (c) Jaccard similarity @ 5%, and (d) top-$k$ overlap ratio @ 5%. The colormap indicates the average value of each measure across all datasets, and we annotate the standard deviation across datasets in each cell.
• Figure 4: Correlation between hyperedge centrality and importance measures in terms of (a) Pearson correlation coefficient, (b) Spearman rank correlation coefficient, (c) Jaccard similarity @ 5%, and (d) top-$k$ overlap ratio @ 5%. The colormap indicates the average value of each measure across all datasets, and we annotate the standard deviation across datasets in each cell.
• Figure 5: Computation time of node and hyperedge measures with respect to different structural quantities in log-log scale. Each subfigure reports the total computation time with respect to (\ref{['fig:runtime:node_node']}) & (\ref{['fig:runtime:edge_node']}) the number of nodes, (\ref{['fig:runtime:node_edge']}) & (\ref{['fig:runtime:edge_edge']}) the number of hyperedges, and (\ref{['fig:runtime:node_degree']}) & (\ref{['fig:runtime:edge_degree']}) the total degree sum, for node and hyperedge measures, respectively.