Hypergraph Centrality Measures

• Hypergraph centrality measures quantify the importance of nodes by capturing higher-order interactions beyond simple pairwise connections.
• They are classified into structural, functional, and contextual methods, each offering specific insights into topology, dynamics, and feature integration.
• Applications span biological, social, and information networks, employing techniques like tensor eigenvector, uplift models, and nonlinear eigenproblems.

A hypergraph centrality measure quantifies the importance of nodes (and sometimes hyperedges) within a hypergraph, a combinatorial structure generalizing graphs by allowing edges to join any number of nodes. Unlike classical graphs, whose centrality frameworks are grounded in tools such as the adjacency matrix and dyadic walks, hypergraphs encode polyadic (higher-order) relationships and thus require substantially richer and more nuanced definitions of centrality. A rapidly growing body of literature, reflected in both comprehensive surveys and methodology papers, has developed a taxonomy of hypergraph centralities comprising structural, functional, contextual, tensor-based, and nonlinear spectral approaches, together with domain-specific applications in biological, social, and information networks.

1. Taxonomy and Classifications of Hypergraph Centrality

A unified taxonomy for hypergraph centrality divides measures into three principal categories: structural, functional, and contextual (Chun et al., 27 Nov 2025).

• Structural (topology-based): Measures relying solely on hypergraph combinatorics, such as degree, path- and walk-based, and subhypergraph-based centralities.
• Functional (dynamics-based): Measures quantifying an entity's impact on global system properties—typically via effect on diffusion, robustness, or coalition value (e.g., higher-order coverage, perturbation-based, Von Neumann entropy).
• Contextual (feature-based): Measures integrating extrinsic data, such as learned embeddings or attribute information, often via attention models or supervised frameworks.

This taxonomy is motivated by the observation that hypergraphs, by virtue of their higher-order structure, require centrality notions capturing not only local connectivity but also systemic influence and feature-driven roles.

Empirical assessment over a suite of 39 measures indicates that while global correlations among node-based scores are moderate (mean Spearman ≈ 0.77), correlations among hyperedge-based scores are significantly lower (mean Spearman ≈ 0.43), reflecting the diverse and often weakly overlapping perspectives of different centrality classes (Chun et al., 27 Nov 2025).

2. Structural Centralities: Degree, Path, Walk, and Vector Measures

Degree centrality remains the most basic structural metric. For a node $v$,

$$C_{\text{deg}}(v) = \deg(v) = |\{ e \in E : v \in e \}|$$

(Chun et al., 27 Nov 2025).

Extensions include variants based on the star expansion or clique (2-section) shadow, yielding path-based measures (such as closeness): $$C_{\text{cls}}(v) = \left( \sum_{u \neq v}\delta_S(v,u) \right)^{-1}$$ where $\delta_S$ is the shortest path in the star expansion (Chun et al., 27 Nov 2025).

Vector centrality assigns to each node a vector $c_i = (c_{i2},\ldots,c_{iD})$ whose $k$-th entry aggregates the eigenvector centralities assigned to hyperedges of size $k$ containing $i$. This vector-valued measure strictly generalizes scalar projections (e.g., clique expansion) and separates the importance of nodes across different interaction orders. Uniqueness and normalization are inherited from the principal eigenvector of the line graph adjacency $A$ (Kovalenko et al., 2021). A plausible implication is that order-sensitivity is essential for distinguishing node roles in real-world hypergraphs with variable edge sizes.

3. Spectral and Nonlinear Eigenvector-Based Centralities

Spectral methods extend eigenvector centrality to hypergraphs via tensor eigenpairs and nonlinear mutual reinforcement between nodes and edges.

Uniform Hypergraphs: Tensor Methods

For an $m$-uniform hypergraph, the adjacency tensor $\mathcal{T}$ generalizes the adjacency matrix. Three spectral definitions are prevalent (Benson, 2018):

• Clique-motif Eigenvector Centrality (CEC): Reduces to ordinary eigencentrality on the motif adjacency of size-2 faces within hyperedges.
• Z-eigenvector centrality (ZEC):

$\mathcal{T} x^{m-1} = \lambda x$

captures a nonlinear, multilinear influence that is not necessarily unique or easily computable.

• H-eigenvector centrality (HEC):

$\mathcal{T} x^{m-1} = \lambda x^{[m-1]}$

admits unique positive solutions and robust, globally convergent power methods.

HEC and CEC generally exhibit high agreement unless higher-order effects dominate, whereas ZEC can diverge and reveal alternative structural insights. Computation scales as $O(n^m)$ for the full tensor but can be mitigated by exploiting sparsity.

Non-Uniform Hypergraphs: Uplift and Nonlinear Models

For non-uniform hypergraphs, no single tensor suffices. The "uplift" operation introduces auxiliary (phantom) nodes to uniformize edge sizes, preserving spectral structure. Using weighted adjacency tensors with combinatorial adjustments, the unique positive H-eigenvector of the uplifted tensor (UHEC) yields robust centrality scores (Contreras-Aso et al., 2023). This method ensures order-consistent rankings (Kendall's $\tau \approx 0.9-0.99$ across orders) and overcomes the cross-order comparability problem inherent in standard tensor slicing. For graphs that are uniform uplifts of pairwise graphs, Z-eigenvectors recover the Perron eigencentrality of the original adjacency (Contreras-Aso et al., 2023).

Node and edge nonlinear eigenvector centrality generalizes further by introducing a system of coupled, nonlinear eigenproblems: $$\begin{cases} \lambda x = g( B W f(y) ) \ \mu y = \psi( B^{\top} N \phi(x) ) \end{cases}$$ where the functions $f, g, \phi, \psi$ are order-preserving and homogeneous; existence and uniqueness hinge on Perron–Frobenius theory for multihomogeneous maps and the degree product $\rho=|\alpha \beta \gamma \delta|<1$ (Tudisco et al., 2021). Linear special cases reduce to HITS-like models; nonlinear choices interpolate between additive, multiplicative, and max-type influences. The model supports global convergence via a nonlinear power method and enables the simultaneous centrality ranking of nodes and hyperedges—critical in applications such as polyadic protein interaction networks, where the centrality-lethality rule extends to the hypergraph setting (Lawson et al., 4 Jun 2024).

4. Functional Centralities: Dynamics, Influence, and Robustness

Functional measures evaluate the effect of node or edge removal on systemic properties such as diffusion or structural efficiency (Chun et al., 27 Nov 2025).

• Hypergraph Gravity Centrality (HGC): Quantifies influence as

$$G^H(i) = \sum_{j\neq i} \frac{k_i k_j}{(d^H(i,j))^2}$$

where $k_i$ is degree and $d^H(i,j)$ aggregates all s-walk-based path lengths with penalization for higher $s$ (Xie et al., 2022). A truncated, semi-local version (LHGC) balances computation and accuracy, especially in large sparse hypergraphs.

Evaluation employs models such as complex contagion, where hyperedge-based thresholding governs multi-node infection spread, and higher-order network efficiency, assessing the resilience of connectivity under targeted node deletions. HGC and LHGC robustly identify both high-impact spreaders (correlation with real-world influence in empirical datasets) and structurally critical nodes for hypergraph dismantling (maximizing network efficiency loss) (Xie et al., 2022). These outcomes show that pairwise and classical hyperedge-based centralities often misidentify key players, especially as polyadic structure dominates.

• Higher-order Von Neumann entropy centrality and grounded-Laplacian eigenvalue centrality are additional functional metrics evaluating the importance of hyperedges based on entropic and spectral perturbations in associated line- or reduced-laplacian graphs (Chun et al., 27 Nov 2025).

5. Contextual and Hybrid Centralities

Contextual measures blend hypergraph topology with side information from node attributes or learned representations (Chun et al., 27 Nov 2025).

• Structure-and-embedding score combines degree or overlap statistics with embedding-based similarity, optimizing an $\alpha$-weighted tradeoff.
• Hypergraph Attention Score originates from hypergraph neural networks, aggregating learned attention coefficients ($\alpha_{v,e}$) over hyperedges to identify salient nodes in feature-aware settings.

Such hybrid centralities are particularly relevant for recommendation or classification tasks on attributed higher-order data. They expand the interpretive power of centrality by integrating external knowledge with purely structural assessments.

6. Relationships with Graph Centrality and Effects of Reduction

The connection between hypergraph and classical graph centrality is nontrivial. Projection onto the clique (2-section) shadow graph may result in qualitatively different rankings compared to genuine tensor-based or higher-order approaches. For example, in the 8-pleated bow-tie hypergraph $B_8$, the principal tensor eigenvector ranks a pair of right-hand vertices above the central hub ($y(r_1)=y(r_2)>y(c)$), while running eigenvector centrality on the shadow graph reverses this ($x(c)>x(r_1)=x(r_2)$) (Clark et al., 2022). This explicit separation between "hyper-centrality" and "shadow-centrality" demonstrates that information about polyadic structure can be lost, and even the identity of key nodes may flip.

This suggests that choosing the proper centrality measure is crucial when higher-order dependencies are significant, and naive projection may not only dilute but also distort the identification of network hubs.

7. Applications and Guidelines for Measure Selection

Empirical applications span domains such as biological networks (protein hypernetworks), social co-affiliations, multi-modal transportation, and neuroscience (Lawson et al., 4 Jun 2024, Chun et al., 27 Nov 2025). In biological systems, nonlinear eigenvector centrality identifies essential proteins and protein complexes, offering both node and edge classification power. In social and information networks, gravity-based and spectral methods reliably flag influential spreaders and structurally vital actors where classical metrics can fail.

Guidelines for selection:

• Scale/complexity: For large/streaming hypergraphs, use degree or power-iteration-based eigenvector centrality due to O(∑deg) or subquadratic runtime characteristics.
• Dynamics: When influence on diffusion or structural damage is critical, functional measures such as HGC or VN-entropy outperform pure structure-based metrics.
• Feature integration: Use contextual measures for tasks requiring both structure and attribute-driven scoring.
• Order-sensitivity: When information encoded in edge order matters, vectorial or order-sensitive spectral centralities reveal distinctions lost by scalar reductions (Kovalenko et al., 2021, Contreras-Aso et al., 2023).

A plausible implication is that empirical similarity among different measures is modest, so the choice of centrality must be governed by the network structure, computational constraints, and the intended notion of importance.

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In summary, hypergraph centrality is a multidimensional concept, encompassing a range of methodologies suited to the complex, higher-order entanglements observed in modern networked systems. Ongoing research continues to clarify existence, uniqueness, computational scalability, and empirical utility, while case studies highlight the risks of indiscriminate projection and motivate the need for application-aligned centrality choice.