{"id":"32922c32-53d4-43ff-980d-07f219b80b69","arxiv_id":"2607.21094","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"By constraining a GIN-style GNN to polynomial operations, APEX derives a degree bound and uses Gauss–Legendre quadrature to compute exact Aumann–Shapley feature and node attributions with 2^{L-1} evaluations.","lead":"This paper introduces APEX, a framework that pairs a deliberately polynomial graph neural network (PolyGIN) with Gauss–Legendre quadrature so that Aumann–Shapley (Integrated-Gradients-style) attributions are computed exactly with a fixed number of evaluations. It is worth reading because it shows how architecture choice can remove the quadrature-error/compute trade-off in path-integral explanations, at the price of giving up non-polynomial GNN components.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fidelity comparison mixes backbones: APEX on PolyGIN vs. baselines on GIN (per Appendix D) — the claimed fidelity advantage may stem from architecture, not exact quadrature.","rationale":"The reader's strongest_claim—the exact quadrature theorem—is correct. The proof of Theorem 1 is a standard degree-doubling induction: linear operations (aggregation, W1, W2, PolyScaleNorm) do not increase degree, and the polynomial activation z + θ⊙z² exactly doubles the degree bound. Substituting the affine path γ(τ) preserves the degree bound of the partial derivative (≤2^L−1), and Gauss–Legendre with m=2^{L−1} points integrates degree ≤2m−1 exactly. The completeness of node-level aggregation (Eq. 12) is immediate. I find no internal flaw in this argument. The reader's weakest_assumption identified the domain restriction to polynomial architectures and logit attribution; this is a valid scope limitation but not a flaw, since the paper explicitly acknowledges it in the Conclusion. However, the more load-bearing concern for the paper's broader central claim is the empirical fidelity evaluation. The paper's abstract promises 'higher attribution fidelity than the compared baselines,' but the main comparison appears to mix backbones, as confirmed by the appendix's same-backbone control showing IG matches APEX on PolyGIN. This does not invalidate the theoretical contribution, but it undermines the empirical demonstration of practical advantage. The reader's CONDITIONAL verdict already requires clarifying baselines and providing same-backbone comparisons; my analysis reinforces that this is the key unresolved issue. I therefore do not change the verdict, but I emphasize that without a same-backbone comparison in the main text with error bars, the fidelity claim is unsupported.","tokens_in":19592,"tokens_out":15538,"duration_ms":142251,"concrete_test":"Reproduce Figures 3 and 4 with all explainers applied to the same trained PolyGIN models for all five datasets, and separately to the same trained GIN models where applicable. Report Fidelity+ and Fidelity− with standard errors over 10 seeds. If APEX's advantage over IG on PolyGIN is within noise, the headline fidelity claim must be revised; if APEX still outperforms on a common backbone, the claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim—that the PolyGIN logit is a degree-2^L polynomial and that m=2^{L−1} Gauss–Legendre points exactly integrate the Aumann–Shapley path—is mathematically sound. The proof in Theorem 1 is a straightforward induction, and the quadrature bound follows from the classical Gauss–Legendre exactness condition. However, the paper's headline empirical claim—'the complete APEX framework achieves higher attribution fidelity than the compared baselines'—rests on a confounded comparison. Section 4.3 does not specify the backbone on which each baseline is evaluated; Table 1 only reports GIN vs. PolyGIN accuracy. Appendix D (Figures 6–7) reveals that when all explainers are evaluated on the same trained PolyGIN models, numerical IG achieves fidelity 'nearly identical' to APEX, while other baselines do not improve. This strongly implies that in the main fidelity figures (Figures 3–4), APEX is applied to PolyGIN while the baselines are applied to GIN. Consequently, any observed fidelity advantage could be due to the PolyGIN architecture itself (e.g., smoother, lower-complexity decision surfaces) rather than to the exact Gauss–Legendre attribution. The exactness theorem is unaffected, but the practical relevance of the framework, as advertised in the abstract, is not established by the experiments as reported. The same-backbone control is relegated to an appendix, and the main text does not qualify the comparison as a system-level comparison of (PolyGIN+APEX) versus (GIN+baseline).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes APEX, a co-design framework pairing a polynomial GNN architecture (PolyGIN) with exact Aumann–Shapley attribution. PolyGIN replaces non-polynomial components with linear operations and the polynomial activation σ_poly(z)=z+θ⊙z², so the pre-softmax logit is a multivariate polynomial in the input features. Theorem 1 gives a degree bound of 2^L for the logit, hence a degree bound of 2^L−1 for the path integrand; Gauss–Legendre quadrature with m=2^{L−1} points then evaluates the attribution integral exactly. Feature-level attributions are summed into signed node-level scores with completeness preserved. Experiments on five graph benchmarks compare predictive accuracy, fidelity, completeness error, and runtime. The theoretical derivation is self-contained and the induction in Theorem 1 is correct; the main weakness is the empirical fidelity comparison, which does not clearly control the attributed backbone.","tokens_in":19912,"tokens_out":5629,"duration_ms":64538,"significance":"If the claims hold, the paper contributes a clean and useful result: a class of GNNs for which Integrated-Gradients-style path integrals are evaluated exactly with a fixed, architecture-determined number of forward–backward evaluations. The proof is elementary but rigorous, and the empirical completeness-error experiment (Figure 5) directly confirms the predicted 8-point exactness for a 4-block PolyGIN. The paper also usefully separates algebraic exactness from 1-WL expressivity in Appendix B. However, the practical relevance advertised in the abstract depends on the fidelity experiments, and those experiments are currently confounded by inadequate reporting of the backbone used for each explainer. The scope is also narrower than the title suggests: exactness holds only for polynomial architectures and pre-softmax logits, a limitation the authors do acknowledge in the conclusion.","major_comments":[{"comment":"The main fidelity comparison does not state which trained backbone each explainer is evaluated on. Table 1 reports only GIN vs. PolyGIN accuracy, and §4.3 merely says 'all methods' are evaluated on correctly classified test graphs. Appendix D reveals that when all explainers are evaluated on the same trained PolyGIN models, numerical IG achieves fidelity 'nearly identical' to APEX. This strongly implies that the headline Figures 3–4 compare APEX on PolyGIN against baselines on GIN. If so, the claimed fidelity advantage may be due to the architecture, not to exact Gauss–Legendre quadrature. Please present the same-backbone comparison as the primary fidelity result, or explicitly reframe Figures 3–4 as a system-level comparison of (PolyGIN+APEX) versus (GIN+baseline) and add a table stating the backbone used for every method.","section":"§4.3, Figures 3–4, Appendix D, Figures 6–7"},{"comment":"APEX attributes the pre-softmax logit f_c, but the fidelity metrics are computed on softmax probabilities p_c after masking/retaining nodes. The paper never justifies that a positive attribution to the logit f_c implies a corresponding change in p_c, since p_c depends on all logits through the softmax denominator. The monotonic link may hold approximately in practice but is not guaranteed. This mismatch is load-bearing for the empirical fidelity claim: an exact logit decomposition can rank nodes differently from a probability-based fidelity protocol. Please either report fidelity based on the attributed logit as well, or state and justify the assumption that logit and probability rankings agree under the perturbation protocol.","section":"§3.1, §4.3, Eqs. (10), (13), (14)"}],"minor_comments":[{"comment":"The title and abstract claim 'exact Aumann–Shapley attribution in GNNs,' but the exactness is conditional on the polynomial architecture and on attributing pre-softmax logits. The conclusion states this explicitly. Please qualify the title/abstract, e.g., 'in polynomial GNNs' or 'under polynomial architectures.'","section":"Title and Abstract"},{"comment":"The notation 'm=2^{L−1}' and the degree bound '2^L−1' are visually confusable in plain text, and several sentences render '2 L−1' without superscripts. Use consistent typesetting, e.g., m=2^{L-1} and deg = 2^L-1.","section":"§4.4, Eq. (9)"},{"comment":"No variance/error bars are shown for the fidelity results, even though Table 1 reports ten seeds. State whether the fidelity curves are averages over seeds and report standard deviations or at least a seed count.","section":"Figures 3–4"},{"comment":"The Mutagenicity label mapping in Table 3 (Label 1 = Non-mutagenic, Label 0 = Mutagenic) is the opposite of the common convention for this dataset. Please verify the encoding and explicitly define the labels used in the visualization; otherwise the qualitative discussion of 'mutagenic' vs. 'non-mutagenic' examples may mislead readers.","section":"Table 3"},{"comment":"The same-backbone control is important and currently appears only in an appendix. Move a condensed version into the main text or at least add a sentence in §4.3 stating explicitly that a same-backbone control is provided and what it shows.","section":"Appendix D"},{"comment":"There is a duplicated phrase in the final sentence of the completeness-error discussion: '...consistent with floating-point roundoff rather than quadrature truncation error, rather than from approximation error...' Remove the redundancy.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The theoretical contribution is sound and likely publishable. The main obstacle is that the central empirical claim is undermined by an unclear/confounded fidelity comparison. The same-backbone experiment in Appendix D is actually a strong consistency check for the method, but it is not integrated into the main narrative. The authors should be asked to make the experimental protocol fully transparent and to either justify or adjust the logit/probability fidelity mismatch."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before reading. The mathematical core is correct: for a GNN built out of linear and quadratic-polynomial blocks, the pre-softmax logit has degree at most 2^L in the input features, which means the Aumann-Shapley integrand has degree at most 2^L-1, and Gauss-Legendre with 2^{L-1} points integrates exactly. The induction in Theorem 1 is clean, the quadrature budget follows from the classical exactness condition, and the completeness derivation in Appendix C is a nice identity. The paper also does a thoughtful job in Appendix B separating the algebraic degree guarantee from WL-style injectivity, and it honestly flags in the conclusion that the exactness guarantee does not extend to ReLU, batch norm, softmax outputs, or attention. This is a genuinely new combination of polynomial architecture and exact attribution.\n\nThe soft spot is the experiments. The main fidelity figures (3-4) compare APEX against GNNExplainer, PGExplainer, GradCAM, FlowX, and IG, but the main text never states which backbone each baseline runs on. The paper's own Appendix D reveals the likely answer: when all explainers are evaluated on the same trained PolyGIN models, numerical IG gives nearly identical fidelity to APEX, while the other baselines do not improve. That is a strong hint that the main comparison is APEX on PolyGIN versus the other methods on standard GIN. So the reported fidelity advantage is a property of the co-designed (PolyGIN + exact-IG) system, not of the exact quadrature per se. The same-backbone control should have been in the main text, not buried in an appendix. The fidelity curves also lack error bars, and no code or data is released.\n\nIs this fatal? I do not think so. The central theoretical claim holds, the architecture is a legitimate contribution, and the paper is transparent about its own limitations. But the abstract and conclusion overstate the empirical case. A reader should treat the fidelity results as system-level, not as evidence that exact quadrature alone makes explanations better.\n\nThis paper is for people working on interpretability or co-designed architectures for GNNs. It deserves a serious referee, with a request to clarify the experimental setup, move the same-backbone comparison into the main text, and add variance estimates or release artifacts. I would read it as a methods paper and would cite the exactness result if I worked on path-attribution methods.","headline":"The exact-integration theory is sound; the headline fidelity result is a confounded system comparison.","tokens_in":759,"tokens_out":791,"would_cite":true,"duration_ms":47265,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that a graph neural network built from polynomial layers can be attributed exactly by Aumann–Shapley path integration with a fixed, architecture-determined number of evaluation points.","keywords":["graph neural networks","Aumann-Shapley attribution","Integrated Gradients","exact quadrature","Gauss-Legendre quadrature","polynomial networks","model-attribution co-design","explainability"],"falsifier":"Train a four-block PolyGIN, compute APEX attributions with the prescribed eight Gauss–Legendre points, and compare against a high-resolution numerical integration (say 512-point Gauss–Legendre or Simpson) on the same model. If the two attribution vectors differ by more than floating-point roundoff, or if the completeness error exceeds machine precision for any input, Theorem 1 fails for that model. A simpler check: insert one ReLU activation into the last block and measure the completeness error of an eight-point rule—it should jump well above FP32 precision.","tokens_in":19440,"feed_emoji":"🧮","tokens_out":5009,"duration_ms":54891,"temperature":0.7,"pith_summary":"This paper argues that the numerical approximation error in path-based attribution of graph neural network predictions can be eliminated by designing the network so that its input-to-logit map is a bounded-degree polynomial. It introduces PolyGIN, a graph-isomorphism-style architecture whose layers use only linear operations and a quadratic polynomial activation, and proves that an L-block PolyGIN has output degree at most 2^L. The derivative along the straight-line baseline-to-input path therefore has degree at most 2^L − 1, so 2^(L−1) Gauss–Legendre points evaluate the Aumann–Shapley integral exactly. Feature attributions aggregate into signed node attributions that sum to the logit difference, giving completeness up to floating-point precision. Experiments on five graph benchmarks show PolyGIN matches a standard graph isomorphism network closely in accuracy while the attribution procedure reaches machine-precision completeness with eight evaluations for a four-block network.","feed_headline":"Polynomial graph nets make Aumann-Shapley attribution exact","feed_subtitle":"With a 4-block model, eight Gauss-Legendre points hit machine-precision completeness.","key_machinery":"PolyGIN, a message-passing network built so that all steps are linear except a quadratic polynomial activation z + θ⊙z²; graph-isomorphism-style sum aggregation, linear projections, and a learnable diagonal scaling are linear, so compositions keep the input-to-logit map polynomial. The load-bearing identity is the Gauss–Legendre exactness property: an m-point rule integrates polynomials of degree up to 2m − 1 exactly. Combined with the degree bound for PolyGIN, this fixes m = 2^(L−1) and eliminates quadrature truncation error.","core_discovery":"The central claim is Theorem 1: for a PolyGIN with L polynomial transformation blocks, the scalar logit is a multivariate polynomial in the input features of degree at most 2^L; hence each partial derivative along the straight-line path has degree at most 2^L − 1. Standard Gauss–Legendre quadrature with m = 2^(L−1) points integrates such a univariate polynomial exactly. The result converts integrated-gradients-style attribution from a resolution-dependent numerical approximation into a fixed finite computation with a deterministic evaluation budget, while additive node-level aggregation (summing feature attributions per node) preserves the completeness identity that the feature attributions","pith_inferences":["The co-design principle could transfer to other continuous attribution frameworks: any path integral whose integrand is polynomial in the path parameter inherits an exact finite quadrature budget, so other baselines or paths may be handled with adjusted degree bounds.","The exponential degree growth (2^L) caps practical depth; the paper uses four blocks, and deeper polynomial networks may need additional scaling or regularization to remain stable.","The exactness guarantee is tied to pre-softmax logits; users attributing probabilities or using batch norm, ReLU, or attention softmax must accept the same quadrature error the framework removes. A testable extension is to design polynomial analogues of attention or normalization that keep the degree bound.","The nearly identical fidelity of numerical integrated gradients and APEX on the same PolyGIN backbone suggests the fidelity gain over other explainers comes from the architecture-plus-attribution combination rather than from exactness alone; the paper argues this in an appendix."],"forward_implications":["For any fixed-depth PolyGIN, exact Aumann–Shapley attribution costs exactly 2^(L−1) forward-backward evaluations, independent of any user-chosen integration resolution.","Completeness holds to floating-point precision: feature attributions sum to the logit difference, and signed node-level attributions decompose a prediction into supporting and opposing evidence.","No convergence sweep or error tolerance is needed; truncation error from path discretization is eliminated under the polynomial assumptions.","The polynomial constraint does not cost much accuracy: on the five benchmarks tested, PolyGIN stays within about one percentage point of a standard graph isomorphism network.","Fidelity experiments (necessary and sufficient subsets) favor the complete framework over several post-hoc explainers on the evaluated datasets."],"fun_headline_variants":["PolyGIN makes Aumann-Shapley attribution exact in GNNs","Exact path integral attribution achieved with polynomial graph nets","APEX: polynomial GNN design for exact feature attribution","From numerical to exact: polynomial GNNs for Aumann-Shapley","Deterministic attribution with polynomial GNNs and Gauss-Legendre"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Every operation that touches the attributed score must be polynomial or linear in the input features, and the attribution must be taken on the pre-softmax logit along a fixed straight-line path from a chosen baseline; introduce a ReLU, batch norm, softmax, or attention softmax anywhere and the degree bound—and with it the exactness guarantee—no longer holds.","fun_headline_variants_meta":{"raw":{"variants":["PolyGIN makes Aumann-Shapley attribution exact in GNNs","Exact path integral attribution achieved with polynomial graph nets","APEX: polynomial GNN design for exact feature attribution","From numerical to exact: polynomial GNNs for Aumann-Shapley","Deterministic attribution with polynomial GNNs and Gauss-Legendre"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1355,"prompt_tokens":790,"completion_tokens":565,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":474}},"tokens_in":534,"tokens_out":565,"duration_ms":6802,"temperature":1.0,"reasoning_tokens":474,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:28:23.391395+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a four-block PolyGIN, compute APEX attributions with the prescribed eight Gauss–Legendre points, and compare against a high-resolution numerical integration (say 512-point Gauss–Legendre or Simpson) on the same model. If the two attribution vectors differ by more than floating-point roundoff, or if the completeness error exceeds machine precision for any input, Theorem 1 fails for that model. A simpler check: insert one ReLU activation into the last block and measure the completeness error of an eight-point rule—it should jump well above FP32 precision.","supporting_citations":[],"review_version":1}