{"id":"10f4e9bf-f030-4a7d-a976-a31e8bb64fcf","arxiv_id":"2607.20962","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Generalizing closeness, betweenness, and eccentricity to a weight-aware hypergraph distance reveals systematic ranking differences vs clique projection and yields a three-class taxonomy of real-world hypergraphs.","lead":"This paper extends three classic network measures—closeness, betweenness, and eccentricity—to hypergraphs using an alternative distance that accounts for hyperedge sizes, overlaps, and weights. It then uses that distance to classify ten real-world higher-order networks into three groups according to which interaction sizes dominate their metric structure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Three-class taxonomy is computed with a single non-unique hypergraph distance; alternative admissible choices may reassign datasets.","rationale":"The reader's weakest assumption identifies the non-unique hypergraph distance as the load-bearing element. I agree. The paper's central claim—that hypergraphs divide into three distinct classes and that the differences between distance choices are systematic—is an empirical generalization. Every measurement used to support it is computed through one specific distance construction with f(x)=1/x. The paper explicitly notes this is a reasonable choice but not the only admissible one, and provides no evidence that the taxonomy, or even the qualitative ranking similarities, would survive under alternative admissible distance formulations. This is not an internal inconsistency; the formulas are well-defined and the code is available. It is a robustness gap. If the proposed concrete test shows class labels are stable across f, the concern is resolved and the conditional verdict can be upgraded. If not, the practical guidance about which interaction orders to retain loses its foundation. I therefore see no reason to alter the reader's conditional verdict: the concern is real, but it is addressable by additional experiments rather than by abandoning the work.","tokens_in":12569,"tokens_out":3149,"duration_ms":34766,"concrete_test":"For each dataset in §4.2.2, recompute the distance vectors d_w,k and d_w using at least two other admissible weight functions, e.g., f(x)=1, f(x)=e^{-x}, and f(x)=1/x^2, while preserving the same distance-construction properties. For each f, assign each dataset to a class using a quantitative convergence criterion (e.g., the smallest k such that the Kolmogorov-Smirnov statistic between the layer-k and full-hypergraph distance distributions is below 0.05). Then compare class labels across f. Also recompute the Kendall/μ curves of Figs. 2-3 for these f. If any dataset changes class or the ranking-similarity trends reverse, the three-class taxonomy and the claimed systematic differences are not robust to the distance definition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All comparisons and the three-class taxonomy rest on the hypergraph distance of Eqs. (1)-(2) with the single weight function f(x)=1/x. The paper itself states in §2.1 that this specific form is 'not necessarily the only one' satisfying the required properties, and that f is 'completely arbitrary.' No robustness check over f is provided. Since the central claim asserts that the observed differences are systematic and that hypergraphs fall into three structurally meaningful classes, a different admissible f could alter distance vectors, centrality rankings, or the layer at which quantile-quantile distributions converge, thereby moving datasets between classes. This would reduce the taxonomy to an artifact of a specific functional choice rather than a property of higher-order topology. The concern is not that the chosen f is wrong, but that the empirical conclusions are underdetermined by the single implementation tested. The paper's own acknowledgement of non-uniqueness, combined with the absence of sensitivity analysis, is the weakest load-bearing point in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes three distance-based topological measures (closeness centrality, betweenness centrality, and node eccentricity) from graphs to weighted hypergraphs using the hypergraph distance of Ref. [16], which accounts for hyperedge sizes, intersections, and weights via a chosen function f. The authors compare these hypergraph measures with their clique-projection counterparts on small illustrative examples and on 10 benchmark hypergraphs plus arXiv tag co-occurrence data. They further introduce a distance-vector representation based on multilayer filtration and use quantile-quantile plots to classify the datasets into three classes: pairwise-interaction dominance, low-order-interaction dominance, and high-order-interaction dominance. The paper concludes that clique projection is often inadequate and that the classification can guide which hyperedge orders to retain.","tokens_in":12797,"tokens_out":5592,"duration_ms":56632,"significance":"If the central claims are robust, the paper provides practical guidance: clique projection can substantially mis-rank nodes in distance-based centralities, and the proposed taxonomy identifies when higher-order layers must be retained. The manuscript is clearly written, the illustrative examples are explicit and tractable, and the source code and data are made available (codeberg), which supports reproducibility. The central limitation is that the entire empirical analysis and taxonomy rest on a single, admittedly arbitrary weight-distance function f(x)=1/x within a non-unique distance definition; this makes the main conclusions conditional until a sensitivity analysis is provided.","major_comments":[{"comment":"The hypergraph distance in Eqs. (1)–(2) depends on the positive function f, and §2.1 states that the specific form is 'not necessarily the only one' and that the weight dependence is 'completely arbitrary'. The paper fixes f(x)=1/x for all comparisons and for the three-class taxonomy in §4.2.2. Since the central claim is that differences between hypergraph and projected distance are systematic and that hypergraphs fall into structurally meaningful classes, this single choice is load-bearing. A different admissible f (e.g., f(x)=x^{-α} with varying α, or a logarithmic form) could alter distance values, centrality rankings, and the layer at which Q-Q distributions converge, potentially moving datasets between classes. Please provide a sensitivity analysis over admissible f to show that the qualitative conclusions—especially the Class 1/2/3 assignments—are robust. Without it, the taxonomy i","section":"§2.1 and §4.2.2"},{"comment":"The assignment of datasets to the three classes is described in words ('effectively equal', 'converge', 'keep changing') and supported by Q-Q plots, but no formal convergence criterion or statistical test is specified. For example, Class 1 is claimed when 'the quantiles of H^{w,2} and H^w are effectively equal'; Class 2 when distributions 'converge for an intermediate layer number k'; Class 3 when quantiles 'keep changing'. Without a precise measure (e.g., maximum absolute quantile deviation below a threshold, or a two-sample test on distance distributions), the classification is not falsifiable and cannot be independently checked. Please define an explicit rule for assigning a dataset to a class and report the numerical values that justify each assignment.","section":"§4.2.2, Figs. 5–8"},{"comment":"The comparison of projected versus hypergraph rankings uses Kendall's rank correlation and the μ_t measure, but no confidence intervals or significance tests are provided. Statements such as 'anticorrelated' (pollinator networks) and 'completely uncorrelated' (pollinator 062) are strong and should be accompanied by uncertainty estimates, especially for datasets with small N (e.g., Enron, Senate). At a minimum, bootstrap CIs for KRC at representative t values, or a permutation test against the null of no association, would support the claim that the observed differences are systematic rather than noise.","section":"§4.1.2, Figs. 2–3"},{"comment":"In the definition of the distance vector (7), components d^{w,k}(i,j) are undefined for k smaller than the layer in which i and j first become connected. The Q-Q analyses in Figs. 5–8 compare distance distributions across layers, but the paper does not state how pairs with undefined entries are treated: are they excluded from the distribution, assigned infinite distance, or handled otherwise? This choice can materially affect whether Q-Q curves 'converge' and thus the class assignment. Please specify the exact subset of node pairs used for each Q-Q comparison and how undefined components are treated.","section":"§4.2, Eq. (7)"}],"minor_comments":[{"comment":"Typo: 'Trough' should be 'Through'. Also 'threedistancebasedtopologicalmeasures' and 'nodepairs' are missing spaces in the abstract.","section":"Abstract"},{"comment":"Typo: 'higher-order interations' should be 'higher-order interactions'.","section":"§3"},{"comment":"The definition of a path π and the indices in the intersection sum are not fully explicit. Please define the length ℓ, the indexing of consecutive hyperedges, and the meaning of the indicator more precisely.","section":"§2.2, Eq. (5)"},{"comment":"The phrase 'quantile-quantile distance relations' is informal; consider using 'quantile-quantile plots of the distance distributions' throughout.","section":"§4.2.2"},{"comment":"The caption contains a typo: 'Relative size the of the largest connected component' should read 'Relative size of the largest connected component'.","section":"Fig. 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper depends substantially on Ref. [16], an in-press/forthcoming work by largely the same group. The authors should ensure that the distance definition and its properties are either fully derived in the present manuscript or clearly stated to be conditional on the acceptance/contents of [16]. Additionally, Ref. [17] already applies the same hypergraph distance to a distance-based measure (global efficiency); the present paper's novelty lies in the centrality/eccentricity extensions and the taxonomy, not in the distance itself. The editor may wish to verify that the overlap with [16] and [17] is appropriately disclosed and that the current manuscript stands on its own."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper does exactly what it advertises: it defines closeness, betweenness, and eccentricity using the hypergraph distance from Ref. [16] (same group), adds a distance-vector via multilayer filtration, and sorts ten real-world hypergraphs plus arXiv into three classes. The genuinely new pieces are the distance-vector construction and the intersection-aware betweenness formula (Eq. 5), which is not just a drop-in distance substitution. The toy examples are explicit and helpful, and they ship code and data, which makes the empirical part reproducible. That is real value for the higher-order networks subfield.\n\nThe soft spots are concentrated around the taxonomic claim. The three classes are assigned on the basis of quantile-quantile plots: the text says the quantiles \"converge\" or \"keep changing\" without any formal criterion, statistical test, or confidence interval. The KRC and μ comparisons are also point estimates with no error bars. None of that is fatal, but it limits how strongly you can state the conclusions.\n\nThe more load-bearing issue is the single distance choice. The paper itself says in §2.1 that the specific form from Ref. [16] is \"not necessarily the only one\" and that f is \"completely arbitrary,\" yet every comparison and every class assignment is computed with f(x)=1/x. Since the central claim is that the observed differences are systematic and that the classes reflect structural features, an alternative admissible f could plausibly shift distance vectors and reassign datasets. The authors explicitly acknowledge non-uniqueness but never test sensitivity. That is the weakest point, and it is real.\n\nI would not call the paper circular: the distance is not fit to the outputs, and the same-group citation is not a problem when the cited work is well-defined and the formula is restated here. But the taxonomy is underdetermined by the single implementation tested.\n\nWho gets value from this? Anyone working on distance-based measures for hypergraphs, especially if they need practical guidance on when clique projection is adequate. It deserves a serious referee: the math is well-defined, the data are public, and the taxonomy is plausible but empirically soft. I would send it to review and ask the authors to add sensitivity analysis over f, formal class-membership criteria, and confidence intervals on the ranking comparisons.","headline":"Useful extension of a self-authored hypergraph distance to three centrality measures, but the three-class taxonomy rests on visual inspection with a single non-unique weight function and needs robustness checks before it can be taken as practical guidance.","tokens_in":13286,"tokens_out":1680,"would_cite":true,"duration_ms":17296,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C82"],"pacs":[],"model":"deepseek-v4-flash","headline":"Distance-based network measures change systematically when computed with a hypergraph-aware distance instead of clique projection, and real hypergraphs sort into three classes by which interaction orders dominate.","keywords":["hypergraphs","distance","centrality","clique projection","higher-order interactions","multilayer representation","eccentricity","betweenness"],"falsifier":"Compute the same centrality rankings, distance-vector quantiles, and class assignments on the same ten datasets using another distance that satisfies the same stated properties, for example with f(x)=1 or f(x)=1/x^2; if the three-class structure or the ranking discrepancies qualitatively disappear or substantially reassign datasets, the paper's central claim would be called into question. Alternatively, find a real hypergraph in the claimed pairwise-dominated class where adding higher-order layers nevertheless changes shortest-path routing or centrality rankings under the paper's own distance.","tokens_in":12499,"feed_emoji":"🕸️","tokens_out":4283,"duration_ms":42108,"temperature":0.7,"pith_summary":"This paper extends three classical distance-based measures—closeness centrality, betweenness centrality, and eccentricity—to weighted hypergraphs using a recently proposed hypergraph distance that accounts for hyperedge sizes, intersections, and weights. Through small illustrative examples and analyses of ten real-world datasets, it demonstrates that the results differ systematically from those obtained via clique projection, and that these differences reflect genuine higher-order structure rather than noise. Building on a multilayer representation of hypergraphs, the authors introduce a distance vector whose components track how distances change as progressively larger interaction orders are included. This lets them classify hypergraphs into three broad types: those dominated by pairwise interactions, those dominated by intermediate interaction orders, and those where all interaction orders matter. The practical payoff is a principled way to decide how many interaction orders must be retained to analyze a system faithfully, thereby reducing computational complexity without losing the information encoded in higher-order connections.","feed_headline":"Three classes of hypergraphs emerge from a size-aware distance","feed_subtitle":"Rankings of closeness, betweenness, and eccentricity shift systematically when hyperedge sizes and weights are counted.","key_machinery":"The key object is the hypergraph distance d_w(i,j) defined in Eqs. (1)-(2), which is computed via a weighted line graph of the hypergraph: each hyperedge becomes a node in the line graph, and the weight between two line-graph nodes depends on the sizes of the corresponding hyperedges, the size of their intersection, and the hyperedge weights through a positive function f. With the choice f(x)=1/x, higher-weight hyperedges contribute shorter distances. This distance is then used to define hypergraph versions of closeness centrality, a betweenness centrality that sums inverse intersection sizes along shortest paths, and eccentricity. A second central object is the distance vector (Eq. 7), obta","core_discovery":"On the paper's own terms, the central claim is that the definition of distance in a hypergraph is not a neutral choice: replacing clique projection with a hypergraph distance that incorporates hyperedge sizes, intersection sizes, and weights yields systematically different values for closeness, betweenness, and eccentricity, and these differences are traceable to concrete structural features of the hypergraph. Analyzing ten real-world datasets plus scientific preprint metadata, the authors show that hypergraphs fall into three classes according to which interaction orders dominate their metric structure: pairwise interaction dominance, low-order interaction dominance, and high-order interact","pith_inferences":["The three-class taxonomy is likely sensitive to the specific choice of the weight function f: a different admissible f (e.g., one that downplays edge sizes) could reassign some datasets between classes, since the authors themselves note that their chosen form is not the only one satisfying the stated properties.","A testable extension: in systems assigned to the high-order-dominated class, spreading processes or shortest-path-based dynamics should measurably deviate from predictions made on the clique projection; the class assignment could thus be validated dynamically, not just statically.","The distance vector could serve as a feature for representation learning on hypergraphs, letting downstream machine-learning tasks choose the filtration depth automatically rather than by a fixed heuristic.","Datasets such as plant-pollinator networks and the disease network, which show strong higher-order dominance, may have structurally different effective geometries than contact networks, which could affect any analysis that assumes a graph-like metric."],"forward_implications":["If the paper's claim is correct, clique projection is not a safe default for distance-based analysis of hypergraphs: for many real systems it yields different, sometimes anticorrelated, rankings of closeness, betweenness, and eccentricity.","The three-class taxonomy gives a practical criterion for deciding how much higher-order structure to retain: for pairwise-dominated hypergraphs, analysis can be restricted to the graph projection with little loss, while high-order-dominated ones require all interaction sizes.","The distance vector can be used as a diagnostic tool: the layer at which quantiles of distance distributions stop changing indicates the highest interaction order needed for faithful analysis.","Extending these measures to directed hypergraphs is a natural next step, as the current framework only covers undirected structures.","The computational cost of exact hypergraph distances (worst-case O(M^3)) motivates scalable approximations for large-scale applications."],"fun_headline_variants":["Size-aware hypergraph distance yields 3 metric classes","Hypergraph distance counting edge sizes shifts centrality and reveals 3 classes","Forget clique projection: size-aware distance reshapes hypergraph centrality","Three hypergraph classes emerge when distance incorporates edge sizes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Every comparison and class assignment in the paper is computed with a single hypergraph distance formula (the one from Ref. [16] with f(x)=1/x), and if that formula is not representative of admissible hypergraph distances, the systematic differences and the three-class taxonomy could change.","fun_headline_variants_meta":{"raw":{"variants":["Size-aware hypergraph distance yields 3 metric classes","Hypergraph distance counting edge sizes shifts centrality and reveals 3 classes","Forget clique projection: size-aware distance reshapes hypergraph centrality","Three hypergraph classes emerge when distance incorporates edge sizes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000588,"raw_usage":{"total_tokens":2557,"prompt_tokens":666,"completion_tokens":1891,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":1834}},"tokens_in":410,"tokens_out":1891,"duration_ms":14877,"temperature":1.0,"reasoning_tokens":1834,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:52:12.000922+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same centrality rankings, distance-vector quantiles, and class assignments on the same ten datasets using another distance that satisfies the same stated properties, for example with f(x)=1 or f(x)=1/x^2; if the three-class structure or the ranking discrepancies qualitatively disappear or substantially reassign datasets, the paper's central claim would be called into question. Alternatively, find a real hypergraph in the claimed pairwise-dominated class where adding higher-order layers nevertheless changes shortest-path routing or centrality rankings under the paper's own distance.","supporting_citations":[],"review_version":1}