{"id":"aaf60fa4-3ab2-4f59-88ea-22f37e9eac47","arxiv_id":"2412.19993","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"CurvGIB reweights graph edges with a learned, task-guided Ricci curvature inside an information-bottleneck framework and improves node classification on several standard graph datasets.","lead":"Graph neural networks send messages along edges, and this paper proposes CurvGIB, which learns edge weights derived from graph curvature using an information-bottleneck objective, so the network passes the most task-relevant messages. The method reports accuracy gains over prior structure-learning and curvature-based baselines on seven node-classification benchmarks and visualizes the learned edge structure.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2's variational upper bound (Eq. 9) is not derived and is false as stated; the tractable CurvGIB objective in Eq. (16) therefore has no valid IB grounding.","rationale":"The reader's REJECT verdict is well supported. I would locate the decisive failure even more narrowly than the reader's two premises. The surrogate-curvature issue (whether Eq. 6 faithfully approximates Ollivier-Ricci curvature) is real, but one could reinterpret κ_IB as a learned edge weight and still have an empirical method; the missing variational upper bound cannot be repaired that way. The abstract promises a 'tractable IB objective function', and the only derivation of that objective is Proposition 2. Appendix A.2's proof never invokes a variational inequality over Z; instead it swaps the label variable Y into an expression for I(Z|κ;X), producing an RHS of Eq. (9) that is not a standard mutual-information upper bound and can be made negative while the LHS is zero. Thus the inequality is false as stated, not merely unproven. Since Section 4.2 and Algorithm 1 optimize Eq. (16) as though it were the IB objective, the entire 'CurvGIB principle' is unsupported. The empirical results, even if reproducible, would at best support a heuristic edge-reweighting method, not the theoretical claim. I therefore see no reason to change the reader's rejection.","tokens_in":16473,"tokens_out":15965,"duration_ms":148470,"concrete_test":"Independently re-derive a variational upper bound for I(Z;X|κ) using the standard inequality I(Z;X) ≤ E_{p(X)}[KL(p(Z|X)||r(Z))] (Poole et al. 2019), and compare it with Eq. (9). Then test Eq. (9) directly on a toy distribution: take Z independent of X and κ (so LHS = 0), choose Y uniform over four symbols with X a noisy binary function of Y such that H(Y|X) > log 2, and set r(Z|κ) uniform over two symbols. For this distribution the RHS of Eq. (9) equals |Z|( -H(Y|X) + log|Z| ) < 0 while the LHS is zero, so the claimed inequality fails. If the bound cannot be recovered for this example, Proposition 2 is false and the CurvGIB objective in Eq. (16) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that CurvGIB 'advances the VIB principle for Ricci curvature optimization' and that Eq. (16) is a tractable IB objective—depends on Proposition 2 (Eq. 9). Appendix A.2 attempts to upper-bound I(Z|κ;X). Starting from I(Z|κ;X) = ∫ p(Z|κ,X) log[p(Z|X,κ)/p(Z|κ)], the proof replaces the integrand with p(X)p(Y|X,κ) log[p(Y|X,κ)/r(Z|κ)] without any variational argument. This is not a standard upper bound: no KL divergence between p(Z|X,κ) and r(Z|κ) appears, the label variable Y is substituted for the representation Z, and the final ratio mixes a probability over Y with a density over Z. Moreover, the lines immediately preceding actually re-derive the lower bound for I(Z|κ;Y), not an upper bound for I(Z|κ;X). Because Eq. (9) is not established, the subsequent loss IBCurv(Zκ)=Σ(1−κ_IB(i,j))d(z_i,z_j) in Eq. (16) is never derived from the stated IB objective; the 'tractable IB objective function' announced in the abstract and used in Algorithm 1 has no valid theoretical grounding. The interpretability and optimal-transport conclusions therefore rest on an unproved, and in fact false, inequality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes CurvGIB, a framework that combines discrete Ricci curvature with the Variational Information Bottleneck (VIB) to learn edge weights for message passing in GNNs. The central idea is to define an 'IB-Curvature' κ_IB via a differentiable surrogate (Eq. 6, adapted from DeepRicci), use Ricci-flow updates (Eq. 13) to refine the graph structure, and train the model with a bi-level objective (Eq. 16) claimed to be a tractable IB loss. The paper reports node-classification results on seven datasets, robustness experiments under graph noise, and visualizations of learned curvature, concluding that CurvGIB outperforms baselines and discovers task-relevant optimal transport structures.","tokens_in":16765,"tokens_out":3267,"duration_ms":35612,"significance":"If the theoretical derivation were sound, combining information bottleneck with discrete curvature would be a novel and potentially useful direction for structure learning in GNNs, and the interpretability claim about transport structures would be interesting. The paper's empirical scope is substantial: seven datasets, multiple baseline families, and an ablation-style parameter analysis. The inclusion of a public code repository is also a positive feature for reproducibility. However, the core theoretical claim—that Eq. (16) is a valid tractable IB objective—is not established; the proof of the key upper bound is mathematically invalid. Since the paper's central contribution is presented as an information-theoretic framework, the soundness of the theory is load-bearing, and I cannot accept the manuscript in its current form.","major_comments":[{"comment":"The proof of Eq. (9) is invalid. The proof begins with an expression for I(Z|κ;X) but then replaces the integrand with p(X)p(Y|X,κ) log[p(Y|X,κ)/r(Z|κ)], substituting the label variable Y for the representation Z and mixing a probability over Y with a density over Z. No variational inequality or KL divergence between p(Z|X,κ) and r(Z|κ) is derived, so the claimed upper bound does not follow. The lines immediately preceding actually re-derive a lower bound for I(Z|κ;Y), not an upper bound for I(Z|κ;X). Because Eq. (9) is not established, the subsequent identification of Eq. (16) as a 'tractable IB objective function' is unsupported.","section":"Appendix A.2, Proposition 2"},{"comment":"The paper never writes the actual training loss. Eq. (12) is an abstract mutual-information minimization involving I(Zκ;Y) and I(Zκ;X), but no variational loss with q(Y|Z) and r(Z) is specified. Eq. (16) defines only IBCurv(Zκ), not the full CurvGIB objective that would be optimized in Algorithm 1. The algorithm says 'Optimize the first phase with Eq.(12)' and 'Optimize the overall framework with Eq.(16)', but neither equation is a computable loss function. The claim of a 'tractable IB objective function' therefore lacks a concrete instantiation, independent of the specific proof defect in Proposition 2.","section":"Section 4.2, Eq. (12) and Eq. (16)"},{"comment":"The notation and probabilistic model are internally inconsistent. Definition 3 writes I(Z|κ;Y) and I(Z|κ;X), but these are not standard conditional mutual information quantities; the paper also writes I(Zκ;Y) and I(Zκ;X) in Eq. (12), and uses both κ and K for curvature. The stated Markov chain MC_CurvGIB : <Y → Zκ → K> is incompatible with the standard IB chain <Y → X → Z> and does not justify the conditioning on κ in the mutual information terms. This ambiguity makes it impossible to determine which quantity the method actually optimizes, and it obscures the role of labels in guiding curvature.","section":"Section 4.1, Definition 3 and Markov chain"},{"comment":"The interpretability claim is self-referential: κ_IB is learned with direct supervision from labels and then used as message-passing weights, so visualizing these weights as the 'optimal information transport pattern' is expected from the training objective, not a discovered structure. To support the geometric interpretation, the paper would need to validate that κ_IB tracks true Ollivier-Ricci curvature (e.g., by comparing against Eq. 4 on small graphs) and that the learned structure differs meaningfully from a label-supervised edge-weight learner without curvature. Without such a baseline, the qualitative claims in Section 5.3 do not demonstrate that the method finds an optimal transport structure.","section":"Section 5.3, 'Learned Graph Curvature' and Figure 6"},{"comment":"The central premise that Eq. (6) faithfully approximates Ricci curvature is not tested. The Laplacian-smoothed affine transform f(Z) is borrowed from DeepRicci, but the paper provides no evidence that the resulting κ_IB approximates the Ollivier-Ricci curvature defined in Eq. (4) for either the original or the refined graph. Since the Ricci-flow update (Eq. 13) and the geometric interpretation both depend on κ_IB being a curvature, this is a correctness risk. A concrete check would be to compute exact Ollivier-Ricci curvature on small graphs and report the correlation or error of Eq. (6), and to verify that the learned structures are not merely reweighted label-information flows.","section":"Section 4.1, Eq. (6)"}],"minor_comments":[{"comment":"There are numerous typos and inconsistent notations, including 'Prelimilary', 'Olliver-Ricci' (Definition 2), 'DeepRcci', 'secoond', 'devided', 'sensity', 'coverage' for 'converges', 'is is' in Appendix A.1, and the stray 'Edo' in Algorithm 1. The paper also alternates between κ and K, and between Z and Zκ, sometimes within the same equation or sentence.","section":"Throughout"},{"comment":"The caption says 'Underlined: best performance of specific backbones' but the text says 'The suboptimal results of each dataset are underlined'. This is contradictory; please clarify whether underlining indicates best or suboptimal.","section":"Section 5.3, Table 2"},{"comment":"The Karate Club dataset is mentioned in Appendix B.1 but not listed in Table 1 or used in the main experiments; please clarify whether results on Karate Club exist and where they are reported.","section":"Section 5.1, Table 1"},{"comment":"The claimed O(n^2) complexity assumes d ≪ n, but Eq. (14) samples a Bernoulli for every node pair, which is O(n^2) even with low-dimensional features; please state this explicitly and discuss the practical cost on the largest datasets.","section":"Appendix B.2, Time Complexity"},{"comment":"The Gumbel-Softmax relaxation is applied to Bernoulli variables, but the notation 'Ber(π_ij)' and the concrete relaxation formula with log π/(1−π) is standard; please ensure τ and the sampling procedure are described precisely, since the refined graph A* is used in later curvature updates.","section":"Section 4.2, Eq. (14)"}],"recommendation":"reject","confidential_remarks":"The paper may contain a useful heuristic method that works empirically, but the theoretical contribution—the VIB derivation—is not sound, and the missing concrete objective makes the method difficult to reproduce from the text. I would not recommend rejection solely on the empirical side, but the central claim is the IB grounding, and that claim fails. The authors should also be aware that the notation inconsistencies and missing derivations make a revision costly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: the paper combines Ricci curvature with a variational information bottleneck in a way that is new and plausibly useful, but the theoretical core does not hold up. The proof of Proposition 2 in Appendix A.2 is invalid, and without it the claimed IB grounding for the loss in Eq. (16) is not established.\n\nWhat is genuinely new here is the idea of learning edge curvatures through an IB-style objective and using a Ricci-flow update to reweight the graph. Earlier curvature methods like CurvGN, BORF, SDRF, and DeepRicci use static or self-supervised curvature; earlier IB structure-learning methods like GIB, VIB-GSL, and RGIB don't use curvature. The combination is original, and the paper situates the work well in both literatures. The experimental setup is fairly standard, with a reasonable selection of baselines.\n\nThe main problem is Proposition 2. The appendix starts from I(Z|κ;X) and after a few lines replaces p(Z|κ,X) with p(X)p(Y|X,κ), then ends with a ratio involving p(Y|X,κ)/r(Z|κ). That is not a valid upper bound: the KL divergence between p(Z|X,κ) and r(Z|κ) never appears, and Y is substituted for Z. The lines immediately before actually re-derive the lower bound for I(Z|κ;Y), not an upper bound for I(Z|κ;X). So Eq. (9) is not established. Consequently, the loss IBCurv(Zκ) = Σ(1−κ_IB(i,j))d(z_i,z_j) in Eq. (16) is never derived from the stated IB objective. The notation is also inconsistent (κ vs K, Z vs Zκ, conditional vs unconditional mutual information), and the actual training loss is never written down explicitly.\n\nEmpirically, the reported numbers look too good to be trusted without code. Baseline F1 scores for GAT have standard deviations like ±17.8 on Cora, which is implausible for a 10-fold cross-validation; many other baselines also show strangely high variance. The graph-denoising experiment has no error bars or baselines. The code link exists but no commit hash or configuration is provided. These issues don't necessarily kill the empirical direction, but they make it hard to verify the claim of consistent improvement.\n\nWho is this for? A reader working on GNN structure learning who wants to see how curvature and IB could be combined. They'll get a plausible framework but should not rely on the theoretical justification. I'd send it to peer review—the idea is worth engaging and the flaws are specific enough that a revision could fix them—but it needs major work on the theory and evaluation before it can be accepted. If the authors can supply a valid bound or reframe the method as a heuristic, it could be a useful paper.","headline":"A clever curvature-plus-IB combination is undermined by an invalid variational bound; the empirical direction is plausible but the theory is not there.","tokens_in":17346,"tokens_out":2916,"would_cite":false,"duration_ms":27860,"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":"The paper claims that learning edge importance as an information-bottleneck-guided Ricci curvature — rather than directly rewiring or learning graph structure — gives a graph neural network its task-optimal message-passing structure, and…","keywords":["graph neural networks","Ricci curvature","information bottleneck","graph structure learning","optimal transport","message passing","variational inference","node classification"],"falsifier":"On a small graph where exact Ollivier-Ricci curvature can be computed by solving the Wasserstein problem, compare the learned edge weights $K$ with the true curvatures: if upweighted edges do not systematically correspond to the curvature pattern the Ricci-flow update is supposed to reinforce, the claim of curvature-guided transport structure is not supported. A second check is to verify the direction of the inequality in Proposition 2's bound on $I(Z|\\kappa; X)$; if the bound does not actually hold, the tractable IB objective is not a valid variational upper bound.","tokens_in":16202,"feed_emoji":"🕸️","tokens_out":6671,"duration_ms":56024,"temperature":0.7,"pith_summary":"This paper proposes CurvGIB, a framework that treats the edge structure of a graph neural network as something to be learned through geometry: instead of rewiring or re-weighting edges by heuristic rules, it uses the information bottleneck principle to learn an edge-level 'IB-curvature' that tells the network which connections carry task-relevant information. The claim is that optimizing this curvature with a Ricci-flow update and a bi-level variational objective yields the optimal transport structure for the downstream task, improving node classification and making the learned message-passing pattern interpretable. The paper reports that CurvGIB outperforms all baselines on all seven datasets, converges faster than attention- and curvature-based alternatives, and degrades gracefully when edges are added or removed. If right, the framework gives GNN practitioners a principled, task-aware alternative to heuristic graph rewiring.","feed_headline":"Curvature-guided bottleneck lifts graph accuracy on every benchmark","feed_subtitle":"It learns which edges carry task information, lifting accuracy on all seven datasets and resisting graph noise.","key_machinery":"The load-bearing object is the differentiable IB-Curvature, an approximation of Ollivier-Ricci curvature (a graph analogue of classical curvature measuring how easily mass transports across an edge), defined as $\\kappa_{\\mathrm{IB}}(i,j) = 1 - \\frac{[L_\\alpha(A) f(Z)]_i - [L_\\alpha(A) f(Z)]_j}{d(z_i, z_j)}$, where $L_\\alpha(A)$ is the graph Laplacian mass distribution and $f$ is an affine transform of the latent representations, making the curvature differentiable in $Z$. This curvature feeds a discrete Ricci-flow edge-weight update $K^{(l+1)} = (1 - \\kappa_{\\mathrm{IB}}^{(l)}) \\, d^{(l)}(Z_\\kappa, Z_\\kappa^T)$, and the resulting weights are turned into a sampled refined adjacency via a Gumbel-Softmax Bernoulli relaxation. The machinery lets the information bottleneck shape the geometry of message passing, with a bi-level optimization alternating between representation learning and structure refinement.","core_discovery":"CurvGIB's central claim is that the underlying optimal transport structure of a graph — which edges should carry the effective information for a task — can be learned by letting the variational information bottleneck guide a discrete Ricci curvature, rather than by directly learning or rewiring the adjacency matrix. The paper introduces a differentiable IB-Curvature surrogate, updates edge weights through a discrete Ricci flow, samples the refined structure with a Gumbel-Softmax relaxation, and optimizes the whole system with a bi-level objective. It reports consistent accuracy and macro-F1 gains across citation, co-authorship, and co-purchase networks, robustness to random edge deletion and addition on Cora, faster convergence than GAT and BORF, and curvature distributions that it interprets as task-relevant backbone versus local-cluster transport.","pith_inferences":["If the curvature surrogate is faithful, the learned IB-curvature could serve as a diagnostic tool for locating the task-critical backbone edges of any graph, even outside a training setting.","The same bi-level scheme could be applied with Forman curvature or other discrete curvatures, avoiding Wasserstein cost entirely and making the geometric claim easier to test on larger graphs.","A direct test would compare the learned edge weights against exact Ollivier-Ricci curvature on small graphs; the paper does not include such a check, so the geometric interpretation currently rests on the DeepRicci surrogate.","Because the IB objective uses a Lagrangian $\\beta$ tuned per dataset, the method may be sensitive to the amount of label-irrelevant information; a per-edge or per-node adaptive $\\beta$ is a natural next step."],"forward_implications":["Node classification accuracy improves over all tested baselines on Cora, Citeseer, PubMed, CS, Physics, Amazon-Computers, and Amazon-Photos.","CurvGIB learns interpretable edge structures: negative IB-curvature marks backbone transport paths, while positive IB-curvature marks locally dense propagation.","The framework is robust to graph noise: on Cora, accuracy degrades only slightly and stabilizes as 10–50% of edges are randomly removed or added.","Training converges about 2.5 times faster than GAT and 2 times faster than BORF on citation datasets.","The optimization recipe of curvature surrogate, Ricci-flow update, and Gumbel-Softmax sampling can in principle be carried over to graph-level tasks, which the paper identifies as future work."],"supporting_citations":[{"why":"Defines Ollivier-Ricci curvature via Wasserstein distance, the geometric quantity CurvGIB optimizes.","marker":"Ollivier 2009"},{"why":"Supplies the differentiable Laplacian-based approximation of curvature that the paper adopts as IB-Curvature.","marker":"Sun et al. 2023"},{"why":"Provides the variational information bottleneck objective and its DKL-based approximation used for graph learning.","marker":"Alemi, Fischer, and Dillon 2017"},{"why":"Source of the variational lower and upper bounds invoked in Propositions 1 and 2.","marker":"Poole et al. 2019"},{"why":"Gives the Gumbel-Softmax concrete relaxation used to sample the refined adjacency structure.","marker":"Jang, Gu, and Poole 2016"},{"why":"The curvature-aware GNN aggregation scheme that CurvGIB extends with learned IB-curvature weights.","marker":"Ye et al. 2020"},{"why":"The original information bottleneck principle that CurvGIB advances to curvature optimization.","marker":"Tishby, Pereira, and Bialek 2000"},{"why":"The variational-information-bottleneck graph structure learning baseline compared against in the experiments.","marker":"Sun et al. 2021"}],"fun_headline_variants":["Learn optimal edge transport with curvature and information bottleneck","CurvGIB: Ricci curvature guided by information bottleneck improves GNN accuracy","Discrete curvature optimization improves graph representation learning","Information bottleneck for discrete Ricci curvature finds key edges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole geometric interpretation rests on the assumption that the differentiable Laplacian-based expression $\\kappa_{\\mathrm{IB}}$ genuinely tracks the Ollivier-Ricci curvature of the graph, so that the learned edge weights reflect real transport geometry rather than just a label-informed re-weighting of features.","fun_headline_variants_meta":{"raw":{"variants":["Learn optimal edge transport with curvature and information bottleneck","CurvGIB: Ricci curvature guided by information bottleneck improves GNN accuracy","Discrete curvature optimization improves graph representation learning","Information bottleneck for discrete Ricci curvature finds key edges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00057,"raw_usage":{"total_tokens":2697,"prompt_tokens":946,"completion_tokens":1751,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1687}},"tokens_in":562,"tokens_out":1751,"duration_ms":13702,"temperature":1.0,"reasoning_tokens":1687,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:41:16.392436+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a small graph where exact Ollivier-Ricci curvature can be computed by solving the Wasserstein problem, compare the learned edge weights $K$ with the true curvatures: if upweighted edges do not systematically correspond to the curvature pattern the Ricci-flow update is supposed to reinforce, the claim of curvature-guided transport structure is not supported. A second check is to verify the direction of the inequality in Proposition 2's bound on $I(Z|\\kappa; X)$; if the bound does not actually hold, the tractable IB objective is not a valid variational upper bound.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Ollivier-Ricci curvature via Wasserstein distance, the geometric quantity CurvGIB optimizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the differentiable Laplacian-based approximation of curvature that the paper adopts as IB-Curvature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the variational lower and upper bounds invoked in Propositions 1 and 2."},{"cited_title":"Graph Structure Learning with Variational Information Bottleneck","cited_arxiv_id":"2112.08903","evidence_quote":"The variational-information-bottleneck graph structure learning baseline compared against in the experiments."}],"review_version":1}