{"id":"853b5134-8dcc-459d-8b5b-c1ca386b3aa2","arxiv_id":"2506.07026","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper introduces αTEC, a tensor-eigenvector centrality that interpolates between edge-driven and triangle-driven importance scores on connected graphs.","lead":"The paper defines a tunable centrality score for network vertices that blends edge and triangle structure through a parameter α, using the leading eigenvector of a nonnegative tensor. It is a proposal for analysts who want to control whether tight triadic clusters or ordinary link density drive importance rankings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.3 does not support the abstract's connectivity claim: the triangle-deletion experiment lacks degree-matched controls and does not actually test vertex-level αTEC rankings.","rationale":"The central mathematical construction survives scrutiny: for α ∈ (0,1], every edge contributes a directed arc in both directions in the associated digraph of A = αA_E + (1−α)A△, so the digraph of a connected graph is strongly connected and Theorem 2.2 is valid. The Perron-Frobenius theorem then gives the claimed unique positive eigenvector up to scaling, so Equation (2.2) and Definition 2.3 are sound. The paper's genuine vulnerability is empirical. The reader's weakest_assumption correctly identifies the missing control in Section 3.3: deleting three vertices of a triangle is not a fair intervention unless the compared triangles are matched on degree and local density. My concern is essentially the same, sharpened: the experiment ranks triangles, not vertices, and never reports the α used, while the abstract claims a vertex-level connectivity result. Because the theorem and definition are correct but the connectivity claim is overreached, the original CONDITIONAL verdict is appropriate. The paper should either add degree-matched and random baselines, or remove the connectivity claim from the abstract and conclusion. The absent regular-graph condition should also be added or removed. These are addressable revisions rather than fundamental flaws.","tokens_in":9293,"tokens_out":10994,"duration_ms":124324,"concrete_test":"Re-run the C. elegans connectivity experiment with a degree-matched null: for each of △1, △2, and △3 in Table 3, sample at least 10,000 random triangles whose three vertices have the same degree multiset, delete those three vertices from the network, and record the increase in the number of connected components. If the observed 6-component splits are not above the 97.5th percentile of this null, or if triangles selected solely by product of degrees produce the same split, then the claimed connectivity effect of αTEC is not established. Also state explicitly which α value produced the scores used for Table 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing empirical assertion is that 'vertices with higher αTEC rankings have a greater impact on network connectivity', but Section 3.3 and Definition 3.1 test something else. The paper defines a triangle importance score I△i = x_p + x_q + x_r and then, in Table 3, deletes all three vertices of the top-ranked triangles and counts the resulting connected components. This is a triangle-level intervention, not a vertex-level test: it does not compare vertices by their αTEC rank, and no value of α is stated for the scores x used in Table 3. Moreover, the comparison is between two disjoint sets of triangles ranked by two different methods; if the αTEC-selected triangles happen to contain higher-degree or more central vertices, the larger component split can be explained by vertex deletion and local density rather than by the proposed centrality. No degree-matched triangles, random-triangle null distribution, or isolated vertex-deletion series is provided, so the abstract's word 'prove' is unsupported. The Conclusion also promises a regular-graph condition that never appears in the body. The mathematical core of the paper, Theorem 2.2 and the uniqueness of the positive Perron vector, is correct; the weak spot is entirely in the experimental support for the connectivity claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an α-triangle eigenvector centrality (αTEC) for connected graphs. The centrality vector x is defined as the positive eigenvector associated with the spectral radius of the nonnegative tensor A = αA_E + (1−α)A_△, where A_E encodes edges and A_△ encodes triangles. The main theoretical result is Theorem 2.2, which states that for a connected graph and α ∈ (0,1], this tensor is weakly irreducible; combined with the Perron–Frobenius theorem for nonnegative tensors, this gives a unique positive eigenvector and hence nonzero centrality scores for every vertex. Numerical experiments on a small graph, Zachary's karate club, the Lusseau dolphin network, and the C. elegans metabolic network show how rankings vary with α. Section 3.3 defines a triangle importance score I△ and reports that deleting the vertices of top-ranked triangles under I△ splits the C. elegans network into more connected components than deleting top-ranked triangles under the Fiedler-vector-based index Ici of Jiang et al. The abstract claims that the experiments prove that vertices with higher αTEC rankings have greater impact on network connectivity.","tokens_in":9490,"tokens_out":3013,"duration_ms":33943,"significance":"If the theoretical claim is correct, αTEC provides a flexible, global centrality measure with a tunable parameter α that interpolates between edge-based and triangle-based influence, and it guarantees strictly positive scores for all vertices in connected graphs. The proof of weak irreducibility in Theorem 2.2 is simple and correct, and the Perron–Frobenius argument is standard. The paper also explicitly gives the componentwise recurrence (2.2), which is useful for understanding the mechanism. However, the experimental support for the connectivity claim in Section 3.3 is not adequate: the intervention is triangle-level vertex deletion, no value of α is stated for the scores used in Table 3, and no controls or statistical tests are provided. Thus the stronger empirical claim in the abstract and the conclusion is not established. The paper would be more valuable if it added a rigorous vertex-level connectivity experiment with degree-matched or randomized baselines.","major_comments":[{"comment":"The claim that 'we experimentally prove that vertices with higher αTEC rankings have a greater impact on network connectivity' is not supported by the experiments. The experiment in Section 3.3 deletes all three vertices of the top-ranked triangles according to the ad hoc score I△, not individual vertices ranked by their αTEC scores. This is a triangle-level intervention, and it does not compare vertices by their αTEC rank. Furthermore, no value of α is given for the scores x used to compute I△ in Table 3, and no degree-matched triangles, random-triangle null distribution, or isolated vertex-deletion series are provided. The observed difference in connected components (6 vs. 3 or 2) could be due to the selected triangles containing higher-degree or otherwise more central vertices, rather than demonstrating a property of αTEC.","section":"Abstract and Section 3.3"},{"comment":"Definition 3.1 is not derived from the αTEC centrality in a principled way. The score I△i = Σ x_p + x_q + x_r is simply the sum of αTEC scores of the three vertices in the triangle, but no justification is given for why this triangle-level aggregation should be considered 'the importance score' of a triangle or why it should be identified with αTEC. The notation 'Σ_{p,q,r}' is also ambiguous because it does not specify whether the sum runs over the three vertices of △i with a factor, or over all permutations. As written, the formula appears to have three separate terms x_p, x_q, x_r, which duplicates the sum. The manuscript needs a clear definition and a justification before this quantity is used to support the connectivity claim.","section":"Section 3.3, Definition 3.1"},{"comment":"The Conclusion states: 'Furthermore, we provide conditions under which all vertices in a regular graph share identical αTEC scores.' No such conditions or theorem appear anywhere in the body of the paper. This claim should either be substantiated with a new theorem and proof, or removed from the conclusion.","section":"Conclusion"}],"minor_comments":[{"comment":"Table 3 has several formatting and typographical issues: 'T riangle' should be 'Triangle', and the entries list triangles as 'c1 = [56, 153, 217]' etc., but the text refers to deleting 'c1, c2 and c3' and '△1, △2 and △3' without explaining the correspondence between the table rows and the deletions. The paragraph would benefit from a clearer description of what exactly was deleted.","section":"Section 3.3, Table 3"},{"comment":"The text in Section 3.2.1 says 'Figure 4 illustrates all triangles of ZKC network', but the figure is labeled as Figure 3. Later, Section 3.2.2 refers to 'Figure 4' for the degree and triangle distributions of the dolphin network, which is correct. Please renumber or fix the cross-references consistently.","section":"Figure cross-references"},{"comment":"The phrase 'an simple example' should be 'a simple example'. Also, the caption of Figure 1 says 'vertices’s αTEC scores', which should be 'vertices’ αTEC scores' or 'vertex αTEC scores'.","section":"Section 3.1, Table 1"},{"comment":"The sentence 'The critical triangles identified by αTEC play a pivotal role in enhancing network Connectivity.' uses an uppercase 'C' in 'Connectivity' and is not supported by the experiment as described. The wording should be softened unless the experiment is substantially improved.","section":"Section 3.3"},{"comment":"In Equation (2.2), the notation is mostly clear, but the manuscript would benefit from explicitly stating that the sums are over edges {i,j} ∈ E(G) and triangles {i,j,k} ∈ V△, respectively, and that the equality holds componentwise for each i. This is implicit but should be stated for readability.","section":"Equation (2.2)"},{"comment":"The numerical experiments do not describe the algorithm used to compute the tensor eigenvector (e.g., the power method), the convergence tolerance, or the number of iterations. Adding these details would improve reproducibility.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core of this paper is correct and elementary, but the experimental section is too weak to support the abstract's and conclusion's connectivity claims. The authors should either add a proper vertex-level deletion experiment with degree-matched controls and a stated α, or explicitly restrict their claims to the triangle-level observations. The missing regular-graph condition in the Conclusion also needs attention; it could be added as a new theorem or deleted. If the authors can address these load-bearing points, the paper may be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the math is fine, the connectivity claim is not. The core result, weak irreducibility of A = αA_E + (1-α)A_△ for connected graphs, follows immediately from the edge terms and is correct; the Perron-Frobenius theorem for weakly irreducible nonnegative tensors then gives unique positive centrality scores. That part deserves a pass. The αTEC family is a real, if incremental, extension of existing tensor eigenvector centralities—at α=1 it reduces to eigenvector centrality, and intermediate α interpolates between edge and triangle influence. Equation (2.2) is transparent and the ranking behavior described in the ZKC example is plausible.\n\nThe soft spot is Section 3.3. The abstract says the authors 'experimentally prove' that higher αTEC vertices have greater impact on network connectivity, but the experiment defines a triangle importance score I△i = x_p + x_q + x_r and then deletes all three vertices of the top triangles, counting components. That is a triangle-level intervention, not a vertex-level test of αTEC rankings. There are no degree-matched controls, no random-triangle baseline, no series of single-vertex deletions, and no value of α stated for the scores used in Table 3. The result—6 components for αTEC triangles versus 2–3 for the Fiedler-cycle triangles—could easily be driven by the selected triangles containing higher-degree or more centrally located vertices. Also, the conclusion promises a condition under which all vertices in a regular graph have identical scores, but that condition never appears in the body. No code or data are released, so the experiments are not independently checkable.\n\nNone of this breaks the definition. The mathematical contribution stands on its own. But the empirical overreach needs to be fixed: either substantially strengthen the experiment with proper baselines and statistical tests, or rewrite the abstract and conclusion to describe the connectivity result as an observation, not a proof.\n\nWho is this for? Someone working on tensor spectral centralities or looking for a tunable, triangle-aware centrality will find a clean definition and a correct uniqueness proof, plus a cautionary example of how easy it is to overinterpret a three-triangle deletion table. I'd send it to a serious referee, but with a clear request to focus on Section 3.3 and the missing regular-graph condition.","headline":"A correct but modest tensor-centrality extension whose empirical connectivity claim overreaches the reported experiment.","tokens_in":10042,"tokens_out":2070,"would_cite":true,"duration_ms":22809,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C50","05C82","15A69"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces αTEC, a centrality that blends edge and triangle structure into a tensor eigenvector and gives every vertex of a connected graph a unique positive score.","keywords":["centrality","tensor eigenvector","triangle structure","edge structure","Perron-Frobenius theorem","weakly irreducible tensor","network connectivity","C. elegans metabolic network"],"falsifier":"Take the C.elegans metabolic network, rank triangles by αTEC score, and build control triangles whose three vertices have the same degree counts as the top-ranked triangles but are not top-ranked. Delete the control vertices and count connected components. If degree-matched controls split the network into 6 or more components, the claimed connectivity advantage of αTEC triangles fails; if they split into fewer than 6, the claim survives.","tokens_in":9069,"feed_emoji":"🔺","tokens_out":6845,"duration_ms":67873,"temperature":0.7,"pith_summary":"Centrality measures rank vertices in networks, but most use either local counts or a single kind of global structure. This paper introduces α-triangle eigenvector centrality (αTEC), which builds one third-order nonnegative tensor from both edges and triangles and takes the unique positive eigenvector for the tensor's spectral radius as the scores. The single parameter α∈(0,1] shifts the balance from edge-driven to triangle-driven importance; larger α weights edge neighborhoods, smaller α weights triangle neighborhoods. Because connected graphs always produce a weakly irreducible tensor, every vertex receives a positive score, and the ranking responds smoothly to α. The paper reports experiments on synthetic and real networks supporting the claim that high-αTEC vertices and triangles occupy structurally critical positions and affect network connectivity.","feed_headline":"One tunable tensor ranks every vertex by edges and triangles","feed_subtitle":"Connected graphs get unique positive scores; top αTEC triangles split C. elegans into six components.","key_machinery":"The load-bearing object is the α-triangle tensor A=αA_E+(1−α)A_△, a third-order nonnegative tensor. Its edge part has an entry 1 at (i,j,j) whenever {i,j} is an edge, and its triangle part has entry 1/2 at each ordered triple forming a triangle. The associated directed graph of this tensor contains an arc from i to j for every edge, so a connected graph makes the tensor weakly irreducible. The Perron-Frobenius theorem for weakly irreducible nonnegative tensors then supplies the unique positive spectral eigenvector that defines the centrality.","core_discovery":"The central construction is the α-triangle tensor A=αA_E+(1−α)A_△ of a connected graph G, where A_E records edges and A_△ records triangles. Theorem 2.2 asserts that A is weakly irreducible for every α∈(0,1], since the associated directed graph contains a directed arc i→j for every edge {i,j} and G is connected; by the Perron-Frobenius theorem for nonnegative weakly irreducible tensors, the spectral radius ρ(A) has a unique positive eigenvector x up to scaling. That x is the αTEC score vector. In components, ρ(A)$x_i^{2}$=α∑_{ {i,j}∈E(G)}$x_j^{2}$+(1−α)∑_{ {i,j,k}∈V△}x_jx_k, so a vertex's score is a self-consistent blend of its edge-neighbors' squared scores and its triangle-neighbors' products. The paper also analyzes rankings as α varies and compares triangle deletion effects in the C.elegans metabolic network, where top αTEC triangles split the connected network into 6 components versus 3 or 2 for the comparison cycle ranking.","pith_inferences":["The same weak-irreducibility argument should carry over to any connected motif hypergraph built from edge pairs plus higher-order motifs, yielding a family of motif-tuned eigenvector centralities with positive scores on every vertex.","A natural control experiment would delete degree-matched random triangles rather than only the top-ranked ones; if those controls split the network into a comparable number of components, the connectivity result would be attributed to vertex deletion or local density rather than to triangle importance.","Because α is continuous, it can be calibrated to data, for example by maximizing agreement with an external importance ranking, turning αTEC from a descriptive index into a fitted centrality model.","For weighted or directed networks, the construction extends by replacing the 0/1 edge and triangle entries with observed weights, and the positive-eigenvector guarantee would survive as long as the directed support remains strongly connected."],"forward_implications":["Every vertex of a connected graph receives a positive αTEC score, so the measure never zeros out vertices that lie outside triangles.","Setting α=1 recovers the classical eigenvector centrality ranking, making αTEC a continuous interpolation toward triangle-dominated rankings as α approaches 0.","Rank changes under decreasing α track the local triangle environment: vertices inside or adjacent to high-triangle regions rise, while vertices in edge-dense but triangle-poor regions fall.","Deleting the vertices of the top-ranked αTEC triangles in the C.elegans metabolic network splits the connected network into 6 components, compared with 3 or 2 for the Fiedler-vector cycle ranking used as the comparison baseline."],"supporting_citations":[{"why":"Supplies the Perron-Frobenius theorem for nonnegative weakly irreducible tensors and the weak-irreducibility characterization used to prove uniqueness and positivity of the αTEC eigenvector.","marker":"[22]"},{"why":"Defines tensor eigenvalues and eigenvectors in the form used in equation (2.1).","marker":"[34]"},{"why":"Provides the variational approach to tensor eigenvalues that underpins the spectral-radius eigenvector construction.","marker":"[28]"},{"why":"Defines classical eigenvector centrality, the special case recovered when α=1.","marker":"[6]"},{"why":"Defines triangle centrality, the local triangle-based measure that αTEC is compared against and extends.","marker":"[8]"},{"why":"Supplies the Fiedler-vector cycle ranking Ici used as the comparison baseline in the connectivity deletion experiment.","marker":"[25]"},{"why":"Provides an earlier two-steps tensor-eigenvector centrality that motivates using spectral eigenvectors for centrality.","marker":"[38]"}],"fun_headline_variants":["Alpha tunes edge vs triangle influence in new centrality","Tensor centrality blends edges and triangles with tunable α","Alpha-triangle eigenvector centrality gives unique rankings","Tunable α blends edge and triangle scores in vertex ranking","New centrality ranks vertices by edges and triangles via α"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that deleting the three vertices of a high-scoring triangle is a fair comparison across triangles, not a proxy for deleting high-degree vertices or vertices in dense regions; the paper does not test this against degree-matched or random-triangle baselines.","fun_headline_variants_meta":{"raw":{"variants":["Alpha tunes edge vs triangle influence in new centrality","Tensor centrality blends edges and triangles with tunable α","Alpha-triangle eigenvector centrality gives unique rankings","Tunable α blends edge and triangle scores in vertex ranking","New centrality ranks vertices by edges and triangles via α"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000563,"raw_usage":{"total_tokens":2687,"prompt_tokens":979,"completion_tokens":1708,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1632}},"tokens_in":595,"tokens_out":1708,"duration_ms":13134,"temperature":1.0,"reasoning_tokens":1632,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:44:08.096291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the C.elegans metabolic network, rank triangles by αTEC score, and build control triangles whose three vertices have the same degree counts as the top-ranked triangles but are not top-ranked. Delete the control vertices and count connected components. If degree-matched controls split the network into 6 or more components, the claimed connectivity advantage of αTEC triangles fails; if they split into fewer than 6, the claim survives.","supporting_citations":[{"cited_title":"Perron–frobenius theo- rem for nonnegative multilinear forms and extensions","cited_arxiv_id":null,"evidence_quote":"Supplies the Perron-Frobenius theorem for nonnegative weakly irreducible tensors and the weak-irreducibility characterization used to prove uniqueness and positivity of the αTEC eigenvector."},{"cited_title":"Eigenvalues of a real supersymmetric tensor","cited_arxiv_id":null,"evidence_quote":"Defines tensor eigenvalues and eigenvectors in the form used in equation (2.1)."},{"cited_title":"Singular values and eigenvalues of tensors: a variational ap- proach","cited_arxiv_id":null,"evidence_quote":"Provides the variational approach to tensor eigenvalues that underpins the spectral-radius eigenvector construction."},{"cited_title":"Factoring and weighting approaches to status scores and clique identification","cited_arxiv_id":null,"evidence_quote":"Defines classical eigenvector centrality, the special case recovered when α=1."},{"cited_title":"Triangle centrality","cited_arxiv_id":null,"evidence_quote":"Defines triangle centrality, the local triangle-based measure that αTEC is compared against and extends."},{"cited_title":"Searching for key cycles in a complex network","cited_arxiv_id":null,"evidence_quote":"Supplies the Fiedler-vector cycle ranking Ici used as the comparison baseline in the connectivity deletion experiment."},{"cited_title":"The two-steps eigenvector centrality in complex networks","cited_arxiv_id":null,"evidence_quote":"Provides an earlier two-steps tensor-eigenvector centrality that motivates using spectral eigenvectors for centrality."}],"review_version":1}