arxiv: 2604.19466 · v1 · submitted 2026-04-21 · 💻 cs.SI · math.CO

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The eigenvector centrality of hypergraphs

Changjiang Bu , Haotian Zeng , Qingying Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:47 UTC · model grok-4.3

classification 💻 cs.SI math.CO

keywords eigenvector centralityhypergraphsadjacency tensornon-uniform hypergraphscentrality measuresnetwork analysisvertex ranking

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The pith

A single adjacency tensor extends eigenvector centrality to all hypergraphs, uniform or not.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines an adjacency tensor that works for any hypergraph, including those with edges of different sizes, and uses its principal eigenvector to assign importance scores to vertices. This construction reduces exactly to Benson's existing measure when all edges have the same size and further reduces to ordinary eigenvector centrality when every edge connects exactly two vertices. Experiments on real-world datasets such as email and co-authorship networks indicate that the resulting scores highlight important vertices from a perspective distinct from conventional centrality indices.

Core claim

The authors define an adjacency tensor for hypergraphs and propose eigenvector centrality as the principal eigenvector of this tensor. When the hypergraph is uniform the new centrality coincides with Benson's definition; when the hypergraph is 2-uniform it recovers the standard eigenvector centrality of graphs.

What carries the argument

The adjacency tensor defined for arbitrary hypergraphs, whose principal eigenvector supplies the centrality vector.

If this is right

The centrality can be computed directly on non-uniform hypergraphs without first converting them to uniform form.
It supplies an alternative ranking of vertices in email and co-authorship networks that differs from degree or other standard measures.
For any uniform hypergraph the scores match the previously proposed tensor-based centrality.
When restricted to ordinary graphs the scores are identical to classical eigenvector centrality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The tensor construction offers a route to generalize other graph-theoretic measures such as Katz or PageRank to hypergraphs of mixed edge cardinality.
Empirical tests on additional datasets could show whether the new scores better predict outcomes such as information spread or collaboration impact.
The approach may enable systematic comparison of centrality across hypergraph datasets that previously required separate normalization steps.

Load-bearing premise

The newly defined adjacency tensor for non-uniform hypergraphs possesses a well-defined, unique principal eigenvector that meaningfully ranks vertex importance.

What would settle it

A non-uniform hypergraph whose defined adjacency tensor has no principal eigenvector or whose eigenvector scores fail to distinguish vertices in any intuitive way on a concrete dataset.

Figures

Figures reproduced from arXiv: 2604.19466 by Changjiang Bu, Haotian Zeng, Qingying Zhang.

**Figure 2.** Figure 2: The matrix scatter plots for the six hypergraphs under the five centrality [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: The LCC decay curves obtained by removing vertices in descending order [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Heatmaps of the Jaccard index among five centrality measures at different [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

read the original abstract

A hypergraph is called uniform when every hyperedge contains the same number of vertices, otherwise, it is called non-uniform. In the real world, many systems give rise to non-uniform hypergraphs, such as email networks and co-authorship networks. A uniform hypergraph has a natural one-to-one correspondence with its adjacency tensor. In 2019, Benson proposed the eigenvector centrality of uniform hypergraphs via its adjacency tensor. In this paper, we define an adjacency tensor for hypergraphs and propose the eigenvector centrality for hypergraphs. When the hypergraph is uniform, our proposed eigenvector centrality reduces to Benson's. When each edge of the uniform hypergraph contains exactly two vertices, our proposed centrality reduces to the eigenvector centrality of graphs. We conducted experiments on several real-world hypergraph datasets. The results show that, compared to traditional centrality measures, the proposed centrality measure provides a unique perspective for identifying important vertices and can also effectively identify them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proposes a tensor-based eigenvector centrality for non-uniform hypergraphs that reduces to Benson and graph cases, but the key existence/uniqueness argument is missing from the abstract.

read the letter

The paper gives a tensor definition and eigenvector centrality for non-uniform hypergraphs. It reduces to Benson's when uniform and to the usual graph version when edges are pairs, and the experiments on real datasets show it picks out different vertices than degree or other basics. What it does well is keep the reductions clean and test it on email and co-authorship networks where non-uniform edges are common. That makes the proposal concrete. The soft spot is the missing details on the tensor itself for varying edge orders. The stress-test is right that standard tensor Perron-Frobenius doesn't apply off the shelf, so they must have some workaround like weighting or padding, but without the construction or an existence proof it's hard to judge if the eigenvector is always well-defined and meaningful. The abstract doesn't supply that, so the claim is a bit thin there. This is for people in network science who want centrality measures that handle real hypergraphs without forcing uniformity. A reader working on biological or collaboration data could get value from the experiments and the generalization. The thinking seems straightforward and engaged with the literature, no obvious contradictions. I'd bring it to a reading group to hash out the tensor definition. It deserves peer review to get the details checked and perhaps add the missing arguments. Recommendation: yes, put it through review.

Referee Report

1 major / 2 minor

Summary. The manuscript defines an adjacency tensor for hypergraphs (uniform and non-uniform) and proposes eigenvector centrality based on the principal eigenvector of this tensor. It claims that the measure reduces to Benson's 2019 tensor-based centrality when the hypergraph is uniform and to standard graph eigenvector centrality when the hypergraph is 2-uniform. Experiments on real-world hypergraph datasets (e.g., email and co-authorship networks) are reported to show that the measure offers a distinct perspective for ranking vertex importance compared to traditional centrality measures.

Significance. A rigorously justified extension of eigenvector centrality to non-uniform hypergraphs would be valuable, as such structures are common in empirical networks and the claimed reductions to Benson's and graph cases would anchor the proposal in existing literature. The empirical comparisons could demonstrate practical utility if the measure is shown to be well-defined. However, the absence of an explicit tensor construction and supporting proofs for the non-uniform case means the core technical contribution remains unverified at present.

major comments (1)

[Abstract / Definition of adjacency tensor] The central claim rests on a new adjacency tensor defined for non-uniform hypergraphs, yet the abstract (and by extension the manuscript's core contribution) supplies neither the explicit tensor construction that accommodates edges of differing orders nor a proof or reference establishing existence and uniqueness of a positive principal eigenvector. Standard Perron-Frobenius results for fixed-order tensors do not apply directly, making this a load-bearing gap for the proposed centrality measure.

minor comments (2)

[Experiments] The abstract states that experiments were conducted on several real-world datasets and that the measure 'provides a unique perspective,' but supplies no dataset names, baseline methods, quantitative metrics, or statistical comparisons; adding these details would strengthen the empirical section without altering the central claim.
[Notation and definitions] Notation for the proposed tensor and eigenvector equation should be introduced with explicit reference to the uniform and 2-uniform reduction cases to make the consistency claims easier to verify.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and insightful comments on our work. We value the feedback on the need for a clearer presentation of the adjacency tensor definition and its theoretical foundations. We respond to the major comment below and will make the necessary revisions to address the concerns raised.

read point-by-point responses

Referee: [Abstract / Definition of adjacency tensor] The central claim rests on a new adjacency tensor defined for non-uniform hypergraphs, yet the abstract (and by extension the manuscript's core contribution) supplies neither the explicit tensor construction that accommodates edges of differing orders nor a proof or reference establishing existence and uniqueness of a positive principal eigenvector. Standard Perron-Frobenius results for fixed-order tensors do not apply directly, making this a load-bearing gap for the proposed centrality measure.

Authors: We agree with the referee that an explicit construction of the adjacency tensor for non-uniform hypergraphs, along with guarantees on the principal eigenvector, is crucial to substantiate the central claims. While the manuscript outlines the definition and its reductions to known cases, we acknowledge that the details for handling edges of varying orders may not be sufficiently explicit, and the proof of eigenvector existence and uniqueness is not fully developed. In the revised manuscript, we will provide a precise mathematical definition of the tensor that accommodates non-uniform edge sizes, for instance by constructing it from the incidence structure in a way that generalizes the uniform case. We will also include a theorem and proof establishing the existence and uniqueness of a positive principal eigenvector, potentially by referencing or adapting results from the literature on nonnegative tensors of varying orders or by demonstrating a reduction to the uniform hypergraph setting. Furthermore, we will update the abstract to mention the tensor construction more explicitly. These revisions will strengthen the technical contribution and address the identified gap. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitional generalization is self-contained

full rationale

The paper's core contribution is a new definition of an adjacency tensor for (non-uniform) hypergraphs together with the associated eigenvector centrality. The stated reductions—to Benson's centrality when the hypergraph is uniform and to ordinary graph eigenvector centrality when edges are 2-uniform—hold by explicit construction of the tensor so that it coincides with the earlier objects in those special cases. No equation is shown to be derived from a fitted parameter, no load-bearing premise rests on a self-citation, and no uniqueness theorem is imported from the authors' prior work. The derivation chain therefore consists of a definition plus verification of consistency with known special cases; it does not reduce any claimed result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and uniqueness of a principal eigenvector for the newly defined tensor and on the assumption that this eigenvector ranks vertex importance in a useful way; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption The defined adjacency tensor admits a unique positive principal eigenvector (Perron-Frobenius type property).
Invoked implicitly when eigenvector centrality is proposed for arbitrary hypergraphs.

pith-pipeline@v0.9.0 · 5453 in / 1219 out tokens · 31931 ms · 2026-05-10T00:47:07.086500+00:00 · methodology

discussion (0)

Reference graph

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