{"id":"17017905-360c-489b-89f8-3c4f251a08c5","arxiv_id":"1908.02967","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper generalizes graph centrality measures to simplicial complexes using the authors' prior higher-order adjacency framework, but the proposed normalizations contain a binomial counting error.","lead":"Real-world networks often have interactions among more than two agents, and this paper proposes several new centrality measures for the higher-order structures, simplices, that encode those interactions. The measures generalize graph degree, eigenvector, closeness, betweenness, and clustering coefficients to simplicial complexes, but contain a normalization error that breaks the main definitions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Degree centrality normalizations use the wrong binomial counts (Defs. 12, 13, 15, 16); as written the measures exceed 1 on a complete complex.","rationale":"The reader identified exactly the load-bearing weakness: the combinatorial denominators in the degree-based definitions are not the counts of simplices containing the fixed simplex. My independent check of Definition 12 confirms it mathematically; the error is not a matter of convention because it makes the centrality exceed 1 in a complete complex, violating the intended normalized ratio. The error propagates through Definitions 13, 15, and 16, so the degree-centrality component of the paper's central claim fails as written. I did not find a different concern that would replace this one: the walk/distance definitions and the eigenvector centrality are separate and not affected by this specific count, but the degree-based toolkit advertised in the abstract is. The rejection stands; the definitional layer would need to be repaired before the measures can be used.","tokens_in":25539,"tokens_out":13017,"duration_ms":139852,"concrete_test":"Take the complete simplicial complex on f0=6 vertices and a fixed 1-simplex σ. Compute Definition 12 with h=2: enumerate the 3-simplices containing σ (there are C(4,2)=6). The paper's denominator C(f0-q-1, q+h)=C(4,3)=4 would give centrality 6/4=1.5; the correct normalization C(f0-q-1, h)=C(4,2)=6 gives 1. For Definition 13, take q=2, p=1, and q'=2: a fixed edge-face is contained in C(4,1)=4 triangles, whereas the printed denominator summand uses C(4,2)=6. If these computations reproduce the numbers above, the false normalization is confirmed and all degree centralities need correction before use.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The pivotal false premise is the combinatorial normalization. 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): the q+1 vertices of σ(q) are already fixed, so one need only choose the h additional vertices from the remaining f0-q-1 vertices. The paper instead writes C(f0-q-1, q+h), which selects q+h additional vertices and does not count simplices containing σ(q). The same mistake appears in Definition 13, where a fixed p-face is contained in C(f0-p-1, q'-p) simplices of dimension q', not C(f0-p-1, q'). These wrong counts are then used in Definitions 15 and 16 (Mq and Nq) and in the strict-upper formulas. The consequence is not cosmetic: in the complete complex on six vertices, a chosen 1-simplex lies in C(4,2)=6 tetrahedra, but the Definition 12 denominator with h=2 is C(4,3)=4, so the centrality equals 1.5, exceeding the intended maximum of 1. Since these ratios are presented as normalized popularity measures, the false count breaks the central definitional claim; every q>0 degree-based centrality and normalized average degree inherits the error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25855,"tokens_out":9832,"duration_ms":102279,"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":[{"comment":"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.","section":"3.1, Definitions 12 and 13"},{"comment":"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.","section":"3.4, Definitions 27 and 30 and Remark 13"},{"comment":"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.","section":"Section 2 and Section 4"}],"minor_comments":[{"comment":"There are several typos and grammatical issues, e.g. 'sencond', 'traingles', 'lenght', and 'graphs networks'; a careful copyedit is needed.","section":"Abstract and throughout"},{"comment":"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.","section":"2.3, Theorem 2"},{"comment":"The phrase 'generalised 1 distance' uses an unexplained superscript; it should be defined or removed.","section":"3.3, Definition 20"},{"comment":"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.","section":"3.4, Example 3"},{"comment":"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.","section":"Section 3.4, Definition 30"}],"recommendation":"major_revision","confidential_remarks":"The normalization errors are pervasive but mechanical; they can be corrected by replacing the binomial terms with C(f0-q-1,h) and C(f0-p-1,q'-p). The clustering-coefficient claim, however, requires more than a local correction: the abstract and Remark 13 overstate what is proved. If the authors are willing to reframe that contribution, the corrected paper could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: the paper has real new ideas about centrality measures for simplices, but the degree-based ones are built on wrong combinatorial counts, so the central normalization claims fail. The stress-test concern is correct.\n\nWhat's genuinely new: Definition 18 (p-adjacency simplicial eigenvector centrality) extends Estrada-Ross in a natural way, and the maximal simplicial degree centrality in Definition 14 is a useful summary statistic. The walk/distance framework and the associated closeness/betweenness are coherent, and the clustering coefficient in Definition 30 is a thoughtful attempt, even if the 'generalises graph clustering' claim is inflated.\n\nThe soft spots are real and load-bearing. In Definition 12 the denominator C(f0-(q+1), q+h) should be C(f0-q-1, h): you only choose the h extra vertices, not a (q+h)-simplex independently. Same error in Definitions 13, 15, and 16. For a complete complex on six vertices, an edge's h=2 upper-degree centrality comes out 1.5 instead of 1. This affects every degree-based measure and the normalized average degrees. Fixing it requires changing the binomial counts, not just a paragraph. The clustering coefficient doesn't recover the standard graph clustering for vertices (Example 3 shows C(v)=1/5 vs CS(v)=0), so the abstract's claim needs tempering or the definition needs to be revised. The empirical support is in the companion paper [20], and there's no data or code here; I'd want to see it before trusting the 'practical utility' statements.\n\nThat said, the flaws are fixable. The definitions are explicit, the paper is readable, and the framework is not a mess. I would send it out, with instructions that the referee checks the combinatorial normalization carefully before any revision is accepted. I wouldn't cite it in its current form.\n\nFor a reading group: maybe, as a case study in how normalization errors can slip into higher-order network measures.","headline":"A useful framework for simplicial centralities, but the degree-based normalizations use wrong binomial counts and need correction.","tokens_in":26325,"tokens_out":3694,"would_cite":false,"duration_ms":37559,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55U10","62R40","91D30","05C82","91C20","82B43","05E45"],"pacs":["89.75.-k","89.75.Fb","89.75.Hc"],"model":"deepseek-v4-flash","headline":"This paper defines simplicial degree, eigenvector, closeness, betweenness, and clustering centralities that generalize graph centrality to multi-agent interactions.","keywords":["simplicial complexes","centrality measures","higher-order adjacency","simplicial degree","eigenvector centrality","clustering coefficient","topological data analysis","network science"],"falsifier":"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.","tokens_in":25339,"feed_emoji":"🕸️","tokens_out":11018,"duration_ms":111723,"temperature":0.7,"pith_summary":"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.","feed_headline":"Simplicial centrality ranks multi-agent communities, not just nodes","feed_subtitle":"Generalizes graph centrality tools to triangles, tetrahedra, and other many-body interactions.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the higher-order adjacencies, simplicial degrees, and multi-combinatorial Laplacian used to define and compute every centrality in the paper.","marker":"[20]"},{"why":"It defines the earlier simplicial adjacency matrix and eigenvector centrality that Definition 18 generalises to $p$-adjacency.","marker":"[13]"},{"why":"It gives the existing simplicial clustering coefficient from which the paper explicitly separates by introducing a new one.","marker":"[29]"},{"why":"It provides the combinatorial Laplacian and oriented-simplex background underlying the degree and Laplacian computations.","marker":"[15]"},{"why":"It introduces graph eigenvector centrality, the classical notion that the paper extends to the $p$-adjacency setting.","marker":"[8]"}],"fun_headline_variants":["Simplicial centrality goes beyond nodes to rank communities","New centrality measures for higher-order network interactions","From nodes to simplices: centralities for many-body networks","Ranking simplicial communities with generalized centralities","Centrality for triangles and tetrahedra, not just nodes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Simplicial centrality goes beyond nodes to rank communities","New centrality measures for higher-order network interactions","From nodes to simplices: centralities for many-body networks","Ranking simplicial communities with generalized centralities","Centrality for triangles and tetrahedra, not just nodes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1701,"prompt_tokens":1075,"completion_tokens":626,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":548}},"tokens_in":691,"tokens_out":626,"duration_ms":6941,"temperature":1.0,"reasoning_tokens":548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:47.011122+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science","cited_arxiv_id":"1908.02583","evidence_quote":"It supplies the higher-order adjacencies, simplicial degrees, and multi-combinatorial Laplacian used to define and compute every centrality in the paper."},{"cited_title":"Estrada and G.J","cited_arxiv_id":null,"evidence_quote":"It defines the earlier simplicial adjacency matrix and eigenvector centrality that Definition 18 generalises to $p$-adjacency."},{"cited_title":"Maletic and M","cited_arxiv_id":null,"evidence_quote":"It gives the existing simplicial clustering coefficient from which the paper explicitly separates by introducing a new one."},{"cited_title":"Goldberg , Combinatorial Laplacians of Simplicial Complexes , Annandale-on-Hudson, New York (2012)","cited_arxiv_id":null,"evidence_quote":"It provides the combinatorial Laplacian and oriented-simplex background underlying the degree and Laplacian computations."},{"cited_title":"doi:10.2307/270732","cited_arxiv_id":null,"evidence_quote":"It introduces graph eigenvector centrality, the classical notion that the paper extends to the $p$-adjacency setting."}],"review_version":1}