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Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper defines simplicial degree, eigenvector, closeness, betweenness, and clustering centralities that generalize graph centrality to multi-agent interactions.

desk verdict A useful framework for simplicial centralities, but the degree-based normalizations use wrong binomial counts and need correction. read the letter →

arxiv 1908.02967 v2 pith:O424UNXQ submitted 2019-08-08 math.AT math.COphysics.soc-ph

Daniel Hernández Serrano , Darío Sánchez Gómez This is my paper · ORCID
classification math.ATmath.COphysics.soc-ph MSC 55U1062R4091D3005C8291C2082B4305E45 PACS 89.75.-k89.75.Fb89.75.Hc
keywords simplicialcomplexescentralitymeasureshigher-orderadjacencydegreeeigenvectorclusteringcoefficienttopologicaldataanalysisnetworkscience
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish that the standard centrality toolkit of network science—degree, eigenvector, closeness, betweenness, and clustering coefficients—can be lifted from graphs to simplicial complexes, where a link becomes a simplex of any dimension and an interaction can involve any number of agents. The authors define each measure using a generalized notion of adjacency in which two simplices are compared through a shared face or through a larger simplex containing both, and the definitions reduce to ordinary graph centrality when the complex is just a graph. The measures are intended to rank simplicial communities—collaborations, neural assemblies, or coauthorship groups—rather than individual nodes. The practical motivation is that many real networks encode many-body interactions that ordinary edges throw away; a working simplicial centrality theory would let network scientists ask which higher-order groups are central and how information flows through them.

What carries the argument

The load-bearing machinery is the higher-order adjacency calculus from the companion work [20]: $p$-lower, $p$-upper, strict, and maximal $p$-adjacency between simplices of possibly different dimensions, together with the associated higher-order degrees and the $(q,h,h')$-multi combinatorial Laplacian, whose matrix entries compute these degrees. On top of this, the paper introduces maximal $p$-nearness and $(p_1,\dots,p_r)$-walks, which yield a generalized metric $d_p$ defined as the shortest $p$-walk between two simplices. That distance is what carries the new closeness and betweenness definitions, while the $p$-adjacency matrix $A^{(q,p)}$ carries the eigenvector centrality. The clustering coefficient combines the maximal simplicial degree with a link rule between maximal neighbour simplices settled by shared faces or $0$-walks.

What would settle it

In the complete simplicial complex on six vertices, take an edge $\sigma$ and apply Definition 12 with $h=2$: $\sigma$ is contained in $\binom{4}{2}=6$ tetrahedra, yet the denominator $\binom{4}{3}=4$ would yield the centrality $6/4=1.5$, above the maximum of $1$ that a proportion should have. Checking this ratio in any complete complex settles whether the normalization is valid.

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Extended reading notes

Core claim

The central claim is that well-defined simplicial analogues of the main graph centrality indices exist, and that they generalize their graph counterparts in a precise sense. Degree-type centralities (Definitions 10–17) count how many larger or adjacent simplices a given simplex touches, normalized by the number of such simplices that could exist; the $p$-adjacency simplicial eigenvector centrality (Definition 18) scores a $q$-simplex by the principal eigenvector of its $p$-adjacency matrix; a new $p$-walk distance (Definition 20) supports $p$-closeness and $p$-betweenness centralities (Definitions 23–24); and a simplicial clustering coefficient (Definitions 27 and 30) measures the density of links among the maximal neighbour simplices of a simplex, generalising the standard graph clustering of a vertex. The authors' stated purpose is a theoretical toolkit for ranking the relevance of a simplicial community and the quality of its higher-order connections, and for studying information transmission and density in simplicial networks.

Load-bearing premise

All of the normalized centrality definitions rest on a counting premise: that the number of larger simplices that can be built on a given face is the binomial value written in the denominator; if that count is too large or too small, the centrality values are no longer honest proportions and can pass their intended maximum.

Editorial extensions

If this is right

  • If the paper's central claim is correct, real-world datasets such as coauthor networks, legislative cosponsorship, school contacts, drug-abuse warning networks, e-mail networks, and online forums can be ranked by the relevance of their higher-order collaborative groups using the maximal simplicial degree centrality.
  • The $p$-adjacency eigenvector centrality singles out $q$-simplices that are $p$-adjacent to many well-connected $q$-simplices, providing a quality-of-connection ranking rather than only a count of neighbours.
  • The $p$-distance and the resulting $p$-closeness and $p$-betweenness centralities allow information-flow questions to be posed at the level of communities: which simplices lie on the most shortest $p$-walks.
  • The simplicial clustering coefficient gives a density score around a simplicial community, and in the vertex-and-edge case it reproduces the graph clustering coefficient.
  • All of these quantities are computable from the multi-combinatorial Laplacian and the sign functions of the generalized boundary operator, so using them does not require computing homology or persistent homology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same $p$-adjacency machinery could define simplicial versions of PageRank, $k$-core decomposition, or percolation centrality by replacing the vertex adjacency matrix with $A^{(q,p)}$.
  • The authors do not discuss weighted simplices; extending these measures to weighted simplicial complexes would require a separate normalization step, since the current definitions count simplices with multiplicity one.
  • A testable extension not stated in the paper is to compute the maximal simplicial degree centrality on the cited real-world datasets and compare its ranking of triangles with the ranking obtained by flattening each complex to its 1-skeleton; disagreements would show where higher-order structure changes the answer.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops a framework of centrality measures for simplices in simplicial complexes, motivated by network-science applications. It uses the higher-order adjacency and degree formalism of the authors' companion paper [20] to define degree-based centralities for q-simplices, a p-adjacency eigenvector centrality, generalizations of closeness and betweenness via new walk and distance notions, and a simplicial clustering coefficient. The main abstract claim is that these measures allow the relevance of simplicial communities to be quantified, and that the clustering coefficient generalizes the standard graph clustering coefficient.

Significance. If the definitions were correct, the paper would offer a reasonably systematic toolkit for ranking simplices and simplicial communities, and the walk/distance formalism in Section 3.3 is a natural extension of earlier ideas. The paper is, however, purely theoretical: no data or code are supplied, and the claims of practical utility refer to the companion paper [20]. The main degree normalizations contain a systematic combinatorial error that invalidates the stated normalization property, and the clustering coefficient does not, as written, recover the standard graph clustering of a vertex. The framework is therefore not yet a reliable basis for applications.

major comments (3)
  1. [3.1, Definitions 12 and 13] The normalization counts are incorrect. In Definition 12, the number of (q+h)-simplices containing a fixed q-simplex σ(q) in a complete complex on f0 vertices is C(f0-q-1, h), because the q+1 vertices of σ(q) are already fixed and h additional vertices must be chosen. The paper instead uses C(f0-q-1, q+h). Similarly, in Definition 13, a fixed p-face is contained in C(f0-p-1, q'-p) q'-simplices, not C(f0-p-1, q'). This error propagates to the strict-upper formula in Section 3.1, to Mq in Definition 15, and to Nq in Definition 16. The consequence is concrete: in the complete complex on six vertices, a chosen 1-simplex with h=2 lies in C(4,2)=6 tetrahedra, while the stated denominator is C(4,3)=4, so the centrality equals 1.5 rather than lying in [0,1] as intended. Since these ratios are presented as normalized popularity measures, the error breaks the central definitional claim for every q>0 degree-based centrality.
  2. [3.4, Definitions 27 and 30 and Remark 13] The claim that the new clustering coefficient generalizes the standard graph clustering coefficient of a vertex is not established and appears false as stated. In Definition 27, the maximal neighbours of a vertex are the maximal simplices containing v, not the graph neighbours of v. In Figure 5(a), the standard graph clustering coefficient is 1/5, but the simplicial clustering coefficient CS(v) is 0. Remark 13 only constructs an auxiliary graph in which CS(σ(q)) becomes the clustering coefficient of a new auxiliary vertex; this is a definitional identity, not a demonstration that the measure reduces to ordinary graph clustering when K is a graph. The abstract's wording should be revised to describe a variant or a new measure rather than a generalization, or a precise theorem should be supplied.
  3. [Section 2 and Section 4] The paper relies heavily on Theorems 1 and 2 and the empirical results of the companion paper [20], but it does not reproduce the proofs, code, or data needed to verify those inputs. This is acceptable for a continuation paper, but the authors should explicitly state which results from [20] are load-bearing and should make the manuscript more self-contained for readers without access to the companion paper.
minor comments (5)
  1. [Abstract and throughout] There are several typos and grammatical issues, e.g. 'sencond', 'traingles', 'lenght', and 'graphs networks'; a careful copyedit is needed.
  2. [2.3, Theorem 2] The notation in Theorem 2 is very dense and some quantities, such as the sums over q'', l with unspecified ranges, are not fully defined; the reader should not have to guess the ranges from the preceding paragraph.
  3. [3.3, Definition 20] The phrase 'generalised 1 distance' uses an unexplained superscript; it should be defined or removed.
  4. [3.4, Example 3] The text says the links are given by a1 and a2 but earlier identifies a3 as the link between t2 and t3; the labels in the figure and the example should be checked for consistency.
  5. [Section 3.4, Definition 30] The definition of 'linked' relies on counting a 0-walk of distance 2 but the counting convention in Remark 12 is stated informally; a formal definition of how multiple 0-walks are handled would improve precision.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular derivation chain: the centrality measures are definitions built on recalled higher-order degree notions; the only self-citation supports the practical-utility claim, not the formal mathematics.

  1. self citation load bearing [Section 1 (Introduction), paragraph beginning 'In fact, we have already revealed in [20]...'; echoed in Section 4 (Conclusions)]
    "In fact, we have already revealed in [20] the potential practical utility of a simplicial degree centrality measure (which we will define in these notes), since an structural analysis of higher-order connectivity of several real-world datasets ... is performed is made there by studying statistical properties and the degree distributions associated with the new simplicial degrees."

    The paper's stated applications claim, that the new degree centralities reveal rich higher-order connectivity in real-world networks, is supported solely by the same authors' companion paper [20], which is not included, machine-checked, or independently reproduced here. The introduction and the conclusions both defer the empirical demonstration to [20], so the practical-relevance premise reduces to a self-citation. However, the formal definitions and the graph-generalization check (Remark 13) are stated in full and do not depend on [20], so the self-citation is not load-bearing for the mathematical derivation itself.

full rationale

The paper's derivation chain is definitional rather than predictive: Section 2 recalls adjacency and degree notions from the authors' companion paper [20], and Section 3 constructs centrality ratios, eigenvector scores, walk distances, and a clustering coefficient from those notions. I found no place where a claimed output is secretly the input under a different name. The graph-clustering generalization in Definition 30 and Remark 13 is verified by an explicit graph-rewriting construction, not assumed, so it is not circular. The only self-citation that carries content is the repeated assertion that the new degree centralities have demonstrated practical utility in real datasets; that assertion is credited entirely to [20], an overlapping-author preprint not included or independently verified here. That is a load-bearing citation for the applications claim, but not for the formal content. The most serious technical defect in the manuscript is not circularity: Definitions 12, 13, 15, and 16 use incorrect binomial counts for the number of higher-dimensional simplices containing a fixed simplex, e.g., C(f0-q-1, q+h) instead of C(f0-q-1, h), and C(f0-p-1, q') instead of C(f0-p-1, q'-p), so the stated normalizations can exceed 1. A wrong count is a correctness error, not a reduction of the output to the input, and therefore does not raise the circularity score. Overall there is one minor self-citation; the formal centrality definitions are self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central definitions rest on the authors' prior adjacency framework [20], a domain modeling assumption, and standard Perron-Frobenius theory. No free parameters are fitted to data. The most fragile entry is the false binomial count assumed in the normalizations, which is the load-bearing error of the paper.

assumptions (4)
  • standard math Perron-Frobenius theorem holds for the p-adjacency matrix A(q,p).
    Invoked in Definition 18 to define the p-adjacency simplicial eigenvector centrality via the largest eigenvalue and positive eigenvector.
  • ad hoc to paper Theorem 1 and Theorem 2 from the companion paper [20] correctly compute higher-order degrees and adjacencies.
    The paper imports these results without proof and builds all centrality measures on them; see Section 2 and the citation to [20].
  • ad hoc to paper The number of (q+h)-simplices that can contain a given q-simplex among f0 vertices is C(f0-q-1, q+h).
    Used in Definition 12 and Definition 15; this count is false, the correct count is C(f0-q-1, h).
  • domain assumption A simplicial complex is the appropriate mathematical model for many-body interactions in real networks.
    Stated in the abstract and Section 1 as the motivation for the whole framework.

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Cite this review

Pith. "Pith review of Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science." pith.science (2026). https://pith.science/paper/O424UNXQ

@misc{pith2026190802967,
  author       = {Pith},
  title        = {Pith review of: Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O424UNXQ}},
  note         = {Machine review of arXiv:1908.02967}
}
read the original abstract

Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.

Figures

Figures reproduced from arXiv: 1908.02967 by the authors.

Figure 1
Figure 1. Maximal 0-adjacency. t2 that are 0-adjacent to t1 (the edges a1 and a2), they are also being counted in the 0-adjacency degree of t1. That is deg0 A(t1) = 3 because t2, a1 and a2 are 0-adjacent to t1. Nonetheless, since a1 and a2 are 0-adjacent to t1, but they are also faces of another simplex 0-adjacent to t1 (they are faces of t2), then, with the maximal 0-adjacent degree of t1 we are not counting a1 and a2, and t… view at source ↗
Figure 2
Figure 2. Examples of (q, h)-boundary operators where we denote [v0, . . . , vp] = v0···p. We shall denote it by σ (q) i ∼U + τ (p) σ (q 0 ) j . If the signs are different, we say that they are dissimilarly oriented with respect to τ (p) . We shall denote it by σ (q) i ∼U − τ (p) σ (q 0 ) j . (2) Let σ (q) i , σ (q 0 ) j and τ (p) oriented simplices. We define the upper sign of σ (q) i and σ (q 0 ) j with respect to τ (p) as … view at source ↗
Figure 3
Figure 3. General simplicial degree of 1-simplex. cosponsoring congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. Example 1. Let us compute the maximal simplicial centrality degree centrality of the 1- simplex σ defined by the vertices v0 and v1 in [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Maximal p-nearness and walks. Example 2. In [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: Simplicial clustering coefficient of a vertex. Example 3. In [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]
Figure 6
Figure 6. Figure 6: Simplicial clustering coefficient of a 1-simplex. Example 4. Let us compute the simplicial clustering coefficient of the 1-simplex σ defined by the vertices v0 and v1 in [PITH_FULL_IMAGE:figures/full_fig_p032_6.png]

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Forward citations

Cited by 1 Pith paper

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  1. Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science

    cs.SI 2019-08 conditional novelty 5.0 of 10

    The paper introduces dimension-agnostic simplicial degree measures and a multi-combinatorial Laplacian, then uses them to describe higher-order connectivity in 17 real-world networks.

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