{"id":"6d8157ab-c353-457a-a951-b43efc6a988c","arxiv_id":"2608.09397","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A message-passing framework for random clustered networks predicts percolation thresholds from local neighborhood overlap, with a new closure coefficient that flags when the prediction will fail.","lead":"This paper introduces a way to estimate when a network will suddenly become connected, using local patterns of overlapping neighborhoods instead of the full network structure. It adds a simple local diagnostic that says in advance whether the estimate can be trusted, which matters for predicting cascades, epidemic spread, and infrastructure failure.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5) asserts that shell-1 message contributions are mutually independent because they are direct neighbors; in loopy graphs this is unsupported and generally false, so the second-order Eq. (9) rests on an unproved factorization.","rationale":"The strongest claim is about Eq. (9) as a refined threshold prediction. The chain from Eq. (4) to Eq. (6) to Eq. (9) depends on the factorization of A(n). The reader identified exactly this as the weakest assumption, and I agree. The lower-order recovery and the synthetic sweep provide indirect support, but neither supplies evidence for the shell-1 independence factorization; the synthetic sweep varies closure while keeping low-order structure fixed and shows GECC tracks error, which supports the diagnostic claim, not the derivation of Eq. (9). The real-network results are also consistent with the factorization failing exactly when GECC is large, but they do not validate the factorization where GECC is small. Because the missing derivation could in principle be supplied and the concern is an unproved assumption rather than a demonstrated numerical contradiction, the appropriate disposition remains CONDITIONAL. Thus the reader's verdict requires no change.","tokens_in":13764,"tokens_out":4756,"duration_ms":55639,"concrete_test":"For the synthetic networks at small conversion parameter, compute the second-order threshold by evaluating Eq. (4) with A(2) estimated directly from brute-force enumeration of the joint residual cluster-size distribution, and compare this to the factorized Eq. (5) solution of Eq. (9). If the two thresholds differ by more than the simulation error bar at any phi, the shell-1 independence assumption is load-bearing for Eq. (9). A minimal version: on an ensemble whose local unit is a four-cycle d-i-x-j-d, enumerate all percolation configurations and measure E[s_i s_j | Gamma_d^(2)] - E[s_i | Gamma_d^(2)] E[s_j | Gamma_d^(2)]; nonzero covariance directly falsifies the factorization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central tractable result, Eq. (9), is obtained from Eq. (4) only after the factorization in Eq. (5). There the paper states that \"message contributions from the nodes in N1(d) remain mutually independent because they are direct neighbors of the degree-d node.\" This is not a consequence of being direct neighbors: in a graph with short loops, after deleting d two shell-1 neighbors of d can remain connected through a shell-2 node or by an edge between them. Their residual cluster sizes are then correlated, the product over i in N1(d) in Eq. (5) is not exact, and the linearized recursion Eq. (6) and its reduced form Eq. (9) have no rigorous support. GECC measures a different consistency property, namely re-induction of generalized edges by internal node pairs, and therefore cannot certify the missing independence. The Supplemental Material, where the derivation is deferred, is not included, so the assumption is neither proved nor tested in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops an ensemble-level loopy message-passing framework for site percolation on random clustered graphs. The authors define nth-order generalized edges as intersections of nth-order local subgraphs, propose an exact recursion (Eq. 4) that is then linearized and reduced to a second-order equation (Eq. 9) depending on degree, distance, shared-neighbor count, and mean residual size. They introduce the generalized-edge closure coefficient (GECC) as an a priori diagnostic of whether a given approximation order is reliable, and test the framework on synthetic networks generated by a controlled unit-replacement procedure and on four real networks from the SNAP collection. The paper reports that GECC(2) tracks the residual second-order threshold error in the synthetic sweep and classifies the real networks into regimes of adequate and inadequate second-order closure.","tokens_in":13973,"tokens_out":2784,"duration_ms":30707,"significance":"If the central recursions are sound, the paper offers a useful step beyond prescribed-motif and single-realization treatments of loopy percolation, with a tractable second-order equation and a structural diagnostic that is computed without percolation simulations. The lower-order limits correctly reduce to known tree-like (Eq. 12) and triangle-aware (Eq. 11) equations, and the synthetic conversion sweep is a clean controlled test showing co-response between GECC(2) and residual threshold error. The validation is not circular: thresholds are computed from independent Monte Carlo simulations, and GECC is computed from local structure alone. The machine-checkable aspects are limited, however, because the derivation of the central recursions and the algorithmic details are deferred to a Supplemental Material that is not included in the manuscript.","major_comments":[{"comment":"The factorization of A^(n) in Eq. (5) is load-bearing but unsupported. The text claims that message contributions from shell-1 neighbors of the degree-d node 'remain mutually independent because they are direct neighbors,' but in a loopy graph two shell-1 neighbors can be connected by an edge or through a common shell-2 node after the focal node is deleted, so their residual cluster sizes can be correlated. Since Eq. (6) is obtained by linearizing Eq. (4) after this factorization, and Eq. (9) inherits the same assumption, the central second-order result currently rests on an unproved independence claim. The manuscript should either prove the factorization under a stated ensemble condition or test it explicitly, for example by comparing the factorized expression with exact enumeration on small loopy ensembles.","section":"Sec. II B 2, Eq. (5)"},{"comment":"Replacing the residual-size distribution P(m|d,l,y,w) by its mean m^(2)_dlyw in Eq. (8) is another unexamined reduction. The percolation threshold is set by the spectral radius of the message-passing matrix, and replacing random coefficients by their averages inside a spectral-radius condition is not generally exact. The manuscript gives no argument that the threshold is insensitive to fluctuations of m or that the mean-field replacement is controlled. This matters because Eq. (9), the main second-order prediction, uses exactly this averaged quantity.","section":"Sec. II C 2, Eq. (8)"},{"comment":"The derivations of Eq. (4), Eq. (5), Eq. (6), the detailed form of the correlation structure A^(2), and the computational procedure for GECC are all deferred to a Supplemental Material that is not included with the manuscript. A referee cannot verify the central recursions, the claimed recovery of lower-order limits, or the GECC algorithm from the main text alone. The authors should provide the Supplement or move the essential derivations into the main text before the technical claims can be assessed.","section":"Supplemental Material and Sec. II"},{"comment":"The claim that GECC serves as an 'a priori validity certificate' is stronger than what the evidence supports. GECC measures one particular consistency property, namely whether internal node pairs re-induce the same generalized edge, but it does not measure the message independence asserted in Eq. (5), nor can it rule out longer-range correlations that only appear at higher orders, as the Discussion itself concedes. The empirical support is a single synthetic sweep showing an approximately linear relation between GECC(2) and the residual threshold error, plus four real networks. This is suggestive but not a certificate; the wording in Sec. II E 3 ('elevate GECC from a descriptive correlate ... into an a priori validity certificate') should be softened or supported by additional controlled tests, such as multiple synthetic families with different loop structures and a demonstration that small GECC(n) reliably implies small threshold error.","section":"Sec. II D 2 and Sec. II E 3"}],"minor_comments":[{"comment":"The notation '−→' in Eq. (7) is nonstandard and unclear; the intended reduction from graph-dependent variables to ensemble-level features should be stated in words or with a more conventional arrow notation.","section":"Sec. II C 2, Eq. (7)"},{"comment":"The symbol R^(n)_G for the set of generalized-edge labels could be confused with the residual structure R^(n)_d→k defined in Eq. (2); a different symbol would improve readability.","section":"Sec. II D 1, Eq. (13)"},{"comment":"The text says the pale-blue shading 'provides a visual guide' for GECC(2), but the shading is not explained in the caption; it should either be removed or described precisely.","section":"Sec. II E 2 and Fig. 2"},{"comment":"The data-availability statement says the code and processed data 'will be made publicly available upon publication'; for reproducibility, the authors should deposit the code and data at the time of submission, especially since the Supplemental Material is currently unavailable.","section":"Methods and Data Availability"},{"comment":"The phrase 'To our knowledge, this is the first ensemble-level formulation' is a novelty claim that would benefit from a more precise comparison with existing ensemble methods, for example clarifying what distinguishes the generalized-edge construction from the motif-based ensembles of Refs. [21,22,31-37].","section":"Sec. I, Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant problem and the empirical strategy is thoughtful, but the central derivation is currently unavailable in the review package and the key factorization in Eq. (5) is not justified. I would ask the authors to provide the Supplemental Material and to address the independence and averaging assumptions before recommending acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the bottom line. This is a serious paper with a genuine new idea: representing ensemble-level messages on generalized edges and using a closure coefficient (GECC) as an a priori reliability diagnostic. The lower-order limits recover the known tree-like and Serrano-Boguna equations, which is a good sanity check. The synthetic sweep is a controlled mechanism test and shows a clean co-response between GECC(2) and residual threshold error. That is real evidence, and GECC is computed without running percolation, so the validation isn't circular.\n\nThe soft spots are real. The stress-test concern about Eq. (5) lands. The paper claims that message contributions from shell-1 neighbors of the degree-d node remain mutually independent 'because they are direct neighbors.' That doesn't follow in loopy graphs. After deleting d, two shell-1 neighbors can be connected through a shell-2 node or by an edge between them, so their residual sizes can be correlated. The factorization of A(n) is the step that makes the linearized recursion Eq. (6) and the reduced Eq. (9) tractable, and it is asserted rather than proven. The derivation is deferred to Supplemental Material, which is not included. That is a load-bearing gap, not a cosmetic one. GECC measures a related but distinct property (re-induction of generalized edges by internal pairs), so it doesn't certify the independence.\n\nThe real-network test is also weaker than it looks. Three of the four networks have GECC(2) near one and large residual threshold errors; the paper reads that as confirmation of GECC. That is consistent with the diagnostic, but it is not a strong demonstration that the second-order theory predicts thresholds where it should. The one clean success (P2P) is a network with very low GECC(1), so it's arguably the easiest case.\n\nThat said, nothing here looks cooked. The authors are transparent about limitations, including GECC's blind spot for longer-range correlations. The citation pattern is sensible. The main issues are addressable: include the Supplemental Material, prove or directly test the shell-1 independence, and ideally add a few more real networks.\n\nWho should read this? Anyone working on message passing or percolation on clustered networks. It's a legitimate candidate for peer review, not a desk reject. My recommendation: send it to review, but the referee should be asked to scrutinize Eq. (5) and the derivation of Eq. (9) carefully, and the authors should be required to make the SM and code available.","headline":"Genuinely new ensemble-level loopy message-passing framework with a plausible diagnostic, but the central factorization is asserted, not proven, and the derivation sits in an unavailable supplement.","tokens_in":14471,"tokens_out":2346,"would_cite":false,"duration_ms":23737,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B43","05C80"],"pacs":["64.60.aq","89.75.Hc"],"model":"deepseek-v4-flash","headline":"An ensemble-level message-passing scheme based on generalized edges—overlaps of neighborhoods—yields a second-order recursion whose percolation thresholds match simulations on clustered networks, and the newly defined generalized-edge…","keywords":["percolation threshold","highly clustered networks","random graph ensembles","message passing","generalized edge","generalized-edge closure coefficient","loop-induced correlations","site percolation"],"falsifier":"Run site percolation on a random graph ensemble with strong shell-1-neighbor correlations—for example, where each node's direct neighbors form dense shared neighborhoods—while keeping GECC(2) small; if Eq. (9)'s predicted threshold deviates substantially from Monte Carlo, the independence assumption underpinning the recursion is false and closure alone is not sufficient.","tokens_in":1634,"feed_emoji":"🕸️","tokens_out":1959,"duration_ms":67464,"temperature":0.7,"pith_summary":"On clustered networks, short loops break the independence assumption that makes standard message passing tractable, so percolation thresholds are hard to predict from local statistics alone. This paper develops an ensemble-level message-passing framework in which messages travel along generalized edges—overlaps of local subgraphs around pairs of nodes—rather than single edges. The second-order approximation, Eq. (9), retains degree, distance, shared-neighbor count, and residual size, and it produces threshold predictions that track Monte Carlo simulations where tree-like and first-order equations fail. The paper also introduces the generalized-edge closure coefficient (GECC), a structural statistic computed entirely from local information, which indicates in advance whether a given approximation order is reliable. The payoff would be an a priori certificate: from local statistics alone, one can tell when low-order predictions are trustworthy and when higher orders are needed.","feed_headline":"Second-order loops sharpen percolation-threshold predictions","feed_subtitle":"A structural closure coefficient, computed from local statistics alone, says when low-order predictions are reliable.","key_machinery":"The central object is the nth-order generalized edge E_{kd}^{(n)} = $Γ_k^{{(n)}}$ ∩ $Γ_d^{{(n)}}$, the overlap of the order-n subgraphs around two nodes; the message is carried by the subgraph pair ($Γ_k^{{(n)}}$, $Γ_d^{{(n)}}$), split into the generalized edge and the residual structure R_{d→k}^{(n)}. A single structural function $A^{{(n)}}$ absorbs all correlations induced by overlapping generalized edges and factorizes under the assumed independence of shell-1-neighbor messages. The tractable engine is Eq. (9), the second-order recursion ϵ_k = q Σ_{d,y,w} m_{kdyw}^{(2)} g(q,y,w) ϵ_d P(d,y,w|k), where g(q,y,w)=1-(1-$q^{{y-1}}$)^w is the connection probability through at least one occupied shared neighbor and $m^{{(2)}}$ is the mean residual size. The diagnostic $GECC^{{(n)}}$ averages the fraction of internal node pairs that fail to re-induce the same generalized edge; low GECC certifies closure at that order.","core_discovery":"On the paper's own terms, the central discovery is that loop-induced correlations in random clustered graph ensembles can be propagated at the ensemble level by replacing single-edge transmission with generalized edges, defined as intersections of order-n subgraphs around pairs of nodes. The message-passing recursion (Eq. 4) is exact at any order, and linearizing it around the percolation transition yields Eq. (6). Retaining second-order structural features—the distance y within the generalized edge, the number w of shared shell-1 neighbors, and the mean residual size m—produces the tractable recursion Eq. (9), whose threshold prediction beats the zeroth-order (tree-like) and first-order (triangle-aware) equations on synthetic and real networks. The paper further claims that the accuracy of a given order is governed by generalized-edge closure, quantified by GECC: when internal node pairs re-induce the original generalized edge, the order is reliable; when they do not, residual threshold error grows, as demonstrated by a controlled synthetic sweep and by four real networks.","pith_inferences":["If GECC remains small at every tested order, the argument suggests the threshold estimate is converging; a natural testable extension is to compute GECC(n) progressively and stop when it decays, a procedure the authors flag for future work.","Because GECC depends only on local structure, it could be adapted to other message-passing-based quantities on clustered networks, such as epidemic thresholds or influence maximization; the paper names these as future directions.","The synthetic sweep varies only the conversion parameter φ, so the near-linear relation between GECC(2) and the residual threshold gap is evidence for closure as the driver; a complementary test would hold GECC fixed while changing loop density to see whether closure, not loop density, predicts error.","The independence assumption in Eq. (5) is the structural weak point; if shell-1-neighbor messages are correlated, one could try to rescue the framework by conditioning on shared shell-2 neighborhoods, which would be a direct refinement of A^{(n)}."],"forward_implications":["The zeroth- and first-order equations reappear as strict limits of the same generalized-edge framework, so the new scheme is a generalization rather than a separate theory.","On networks with good second-order closure, q_c^{(2)} follows simulated thresholds closely, so structural statistics alone can replace guesswork about low-order reliability.","On networks with GECC(2) close to one, even the second-order prediction remains far from simulation, and GECC identifies these networks before any percolation simulation is run.","GECC is computed purely from local statistics and involves no percolation dynamics, so it can warn in advance that a given truncation order is inadequate and that higher-order or alternative treatments are needed.","Because the framework operates at the ensemble level and does not require a realized adjacency matrix, it applies to large clustered random-graph ensembles described by local statistics."],"supporting_citations":[{"why":"Supplies the progressively refined order-n subgraph perspective that the ensemble-level formulation adapts.","marker":"[39]"},{"why":"Provides the triangle-aware first-order message-passing equations that the framework recovers at n=1.","marker":"[28–30]"},{"why":"Establishes the tree-like message-passing equation that the framework recovers at zeroth order.","marker":"[10]"},{"why":"Models clustered random graphs with prescribed triangles, a baseline that generalized-edge statistics generalize.","marker":"[21]"},{"why":"Gives percolation in random clustered networks, an earlier loopy-threshold baseline compared against here.","marker":"[22]"},{"why":"Supplies the real-world networks used to test GECC and the threshold predictions.","marker":"[42]"}],"fun_headline_variants":["GECC: local gauge for percolation prediction reliability","Ensemble loopy message passing enhances percolation threshold estimates","Second-order loops outperform tree-like percolation equations","Generalized edges enable accurate percolation thresholds on clustered nets","Local closure coefficient diagnoses percolation approximation errors"],"cache_read_input_tokens":16640,"weakest_assumption_plain":"The derivation assumes that messages from the direct neighbors of a node are mutually independent, even though those neighbors can share shell-2 nodes or be directly linked in a clustered graph.","fun_headline_variants_meta":{"raw":{"variants":["GECC: local gauge for percolation prediction reliability","Ensemble loopy message passing enhances percolation threshold estimates","Second-order loops outperform tree-like percolation equations","Generalized edges enable accurate percolation thresholds on clustered nets","Local closure coefficient diagnoses percolation approximation errors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3297,"prompt_tokens":1013,"completion_tokens":2284,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":2204}},"tokens_in":629,"tokens_out":2284,"duration_ms":18135,"temperature":1.0,"reasoning_tokens":2204,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:03:32.763362+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run site percolation on a random graph ensemble with strong shell-1-neighbor correlations—for example, where each node's direct neighbors form dense shared neighborhoods—while keeping GECC(2) small; if Eq. (9)'s predicted threshold deviates substantially from Monte Carlo, the independence assumption underpinning the recursion is false and closure alone is not sufficient.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the progressively refined order-n subgraph perspective that the ensemble-level formulation adapts."},{"cited_title":"Morone and H","cited_arxiv_id":null,"evidence_quote":"Models clustered random graphs with prescribed triangles, a baseline that generalized-edge statistics generalize."},{"cited_title":"Qian, D.-D","cited_arxiv_id":null,"evidence_quote":"Gives percolation in random clustered networks, an earlier loopy-threshold baseline compared against here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the real-world networks used to test GECC and the threshold predictions."}],"review_version":1}