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REVIEW 5 major objections 5 minor 28 references

Discrete Curvature Graph Information Bottleneck

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict A clever curvature-plus-IB combination is undermined by an invalid variational bound; the empirical direction is plausible but the theory is not there. read the letter →

arxiv 2412.19993 v1 pith:DKIYCYUX submitted 2024-12-28 cs.LG

Xingcheng Fu , Jian Wang , Yisen Gao , Qingyun Sun , Haonan Yuan , Jianxin Li , Xianxian Li This is my paper · ORCID
classification cs.LG
keywords graphneuralnetworksRiccicurvatureinformationbottleneckstructurelearningoptimaltransportmessagepassingvariationalinferencenodeclassification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

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.

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 (5)
  1. [Appendix A.2, Proposition 2] 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.
  2. [Section 4.2, Eq. (12) and Eq. (16)] 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.
  3. [Section 4.1, Definition 3 and Markov chain] 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.
  4. [Section 5.3, 'Learned Graph Curvature' and Figure 6] 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.
  5. [Section 4.1, Eq. (6)] 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.
minor comments (5)
  1. [Throughout] 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.
  2. [Section 5.3, Table 2] 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.
  3. [Section 5.1, Table 1] 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.
  4. [Appendix B.2, Time Complexity] 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.
  5. [Section 4.2, Eq. (14)] 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.

Circularity Check

3 steps flagged · score 6.0 of 10

CurvGIB's derivation is partially circular: the curvature variable cancels out of its own objective (Eq. 16 reduces to a Laplacian smoothness term), Proposition 2's 'upper bound' substitutes labels for representations, and the learned 'optimal transport' curvature is fit to the same labels used to interpret it.

  1. self definitional [Section 4.2, Eqs. (10), (13), and (16)]
    "κIB(i, j) = 1− [Lα(A⋆)f (Z)]i − [Lα(A⋆)f (Z)]j / d(zi, zj) (10) ... K(l+1) = (1 − κ(l)_IB) d(l)(Zκ, ZT_κ) (13) ... IBCurv(Zκ) = Σ_{ij} (1 − κIB(i, j)) · d(zi, zj) (16)"

    Substituting the defining expression for κ_IB into Eq. (16) cancels the curvature variable: IBCurv(Zκ) = Σ_{ij} ([Lα(A⋆)f(Z)]_i − [Lα(A⋆)f(Z)]_j). This is a graph Dirichlet energy on f(Z), with no dependence on the learned curvature κ_IB. The quantity announced as the 'tractable IB objective' and as 'curvature optimization' therefore reduces by construction to a smoothness regularizer; the Ricci-curvature object is eliminated from its own optimization objective.

  2. other [Appendix A.2, Eq. (22); used as Proposition 2, Eq. (9)]
    "I(Z|κ;X) = ... ≤ Z p(X)p(Y|(X,κ)) log p(Y|(X,κ))/r(Z|κ) dκdXdZ"

    The left side is the compression term I(Z|κ;X), an integral over the representation Z. The right side, offered as its upper bound, is an expectation over p(X)p(Y|X,κ) of log[p(Y|X,κ)/r(Z|κ)]: the label Y appears in the numerator and Z appears only in the variational denominator. No KL divergence between p(Z|κ,X) and r(Z|κ) is formed; the proof has substituted the target label for the representation. Consequently Eq. (9) is not a bound on I(Z|κ;X) but a relabeled supervised likelihood, and the tractable objective Eq. (16) is not derived from the IB objective Eq. (7). The compression term is defined to contain the very label it was supposed to compress away.

1 more flagged steps
  1. fitted input called prediction [Section 4.1, after Eq. (7); interpreted in Section 5.3 'Learned Graph Curvature']
    "The learning procedure can be expressed through the Markov Chain: MCCurvGIB :< Y → Zκ → K>. IB-Curvature represents the optimal transport pattern of the information most relevant to Y."

    The curvature κ_IB is computed from Zκ via Eqs. (10) and (13), while Zκ is optimized in Eq. (12) with the supervised objective −I(Zκ; Y) + βI(Zκ; X). Thus the learned curvature is a deterministic function of label-trained representations. Presenting the resulting κ_IB as the discovered 'optimal transport pattern of the information most relevant to Y' is a restatement of the training objective, not an independent geometric finding. The interpretability claim is therefore fitted to the same labels that are later used to validate it.

full rationale

This is not a case of pervasive self-citation: the differentiable curvature surrogate is taken from DeepRicci (Sun et al. 2023), whose author list does not overlap with the present authors, and the GCN/GAT/BORF/SDRF baselines are external. The one self-citation, VIB-GSL (Sun et al. 2021), is used only for qualitative comparison in Figure 6 and is not load-bearing. However, the central derivation chain contains three load-bearing reductions. First, Eq. (16) is obtained by substituting the definition of κ_IB into K; algebraically the curvature cancels, leaving Σ([Lα(A⋆)f(Z)]_i − [Lα(A⋆)f(Z)]_j), a Dirichlet energy. The object named 'curvature optimization' does not optimize the learned curvature at all. Second, Proposition 2 is the only bridge from the IB objective Eq. (7) to the tractable loss, but its proof in Appendix A.2 replaces I(Z|κ;X) with an integral over p(X)p(Y|X,κ) log[p(Y|X,κ)/r(Z|κ)], which is not a valid mutual-information upper bound and injects the label into the compression term. Third, the interpretability claim that κ_IB represents the task-relevant optimal transport pattern is a restatement of the supervised objective used to train Zκ. The empirical benchmark numbers are externally compared and are not themselves circular, which keeps the score below 8; but the theoretical 'VIB-advancing curvature optimization' and the 'optimal transport discovery' claims reduce by construction to a supervised smoothness regularizer, warranting a partial-circularity score of 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The framework rests on a pre-existing differentiable curvature surrogate, a heuristic discrete Ricci flow, and variational bounds that are asserted rather than correctly proved. The learned quantities (κ_IB, A⋆) are internal model components, not independent discoveries.

free parameters (4)
  • β (Lagrangian multiplier) = grid-searched per dataset over {1e-1, 1e-2, 1e-3, 1e-4, 1e-5, 1e-6}
    Controls the trade-off between label sufficiency and input compression in Eq. (1)/(7); selected per dataset by grid search.
  • τ (Gumbel-Softmax temperature) = not reported
    Sharpness of the Bernoulli relaxation for structure sampling in Eq. (14); chosen by hand but the value is not stated.
  • α (mass weight in curvature) = not reported
    Fixes the mass distribution in the approximate Ollivier-Ricci curvature (Eq. 3/5); value not stated in the paper.
  • GNN depth l = searched over {2, 4, 6, 8}
    Depth search per dataset reported in Section 5.2.
assumptions (5)
  • standard math Ollivier-Ricci curvature definition (Eq. 4) measures edge transport efficiency.
    Well-established definition from Ollivier (2009), used as the geometric starting point.
  • domain assumption The DeepRicci surrogate curvature (Eq. 6) is a valid differentiable proxy for Ollivier-Ricci curvature.
    Adopted from Sun et al. (2023) without validation in this paper; the whole structure update depends on it.
  • ad hoc to paper Discrete Ricci flow (Eq. 13) evolves edge weights toward an optimal transport structure.
    The update K^(l+1) = (1 - κ_IB) d(·) is a heuristic design choice, not derived from the IB objective.
  • domain assumption Variational bounds in Propositions 1 and 2 are valid.
    The paper claims proofs, but the proof of Proposition 2 in Appendix A.2 is invalid, so this is an unverified assumption for the stated objective.
  • ad hoc to paper Markov chain <Y → Zκ → K> holds for the IB derivation.
    Stated in Section 4.1; the conditioning of mutual information on κ is unusual and not justified.
invented entities (2)
  • IB-Curvature κ_IB
    purpose: An edge weight learned by the IB objective, intended to represent the importance of each edge for transporting task-relevant information.
    Validated only through the same node-classification task used to fit it; no independent geometric benchmark or falsifiable prediction.
  • Optimal information transport structure A⋆
    purpose: A refined adjacency matrix sampled from a Bernoulli distribution with sigmoid(K) probabilities, claimed to be the task-optimal transport structure.
    The optimality claim is self-referential: the structure is optimized for the task and then presented as the discovered optimal structure.

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Pith. "Pith review of Discrete Curvature Graph Information Bottleneck." pith.science (2026). https://pith.science/paper/DKIYCYUX

@misc{pith2026241219993,
  author       = {Pith},
  title        = {Pith review of: Discrete Curvature Graph Information Bottleneck},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKIYCYUX}},
  note         = {Machine review of arXiv:2412.19993}
}
read the original abstract

Graph neural networks(GNNs) have been demonstrated to depend on whether the node effective information is sufficiently passing. Discrete curvature (Ricci curvature) is used to study graph connectivity and information propagation efficiency with a geometric perspective, and has been raised in recent years to explore the efficient message-passing structure of GNNs. However, most empirical studies are based on directly observed graph structures or heuristic topological assumptions and lack in-depth exploration of underlying optimal information transport structures for downstream tasks. We suggest that graph curvature optimization is more in-depth and essential than directly rewiring or learning for graph structure with richer message-passing characterization and better information transport interpretability. From both graph geometry and information theory perspectives, we propose the novel Discrete Curvature Graph Information Bottleneck (CurvGIB) framework to optimize the information transport structure and learn better node representations simultaneously. CurvGIB advances the Variational Information Bottleneck (VIB) principle for Ricci curvature optimization to learn the optimal information transport pattern for specific downstream tasks. The learned Ricci curvature is used to refine the optimal transport structure of the graph, and the node representation is fully and efficiently learned. Moreover, for the computational complexity of Ricci curvature differentiation, we combine Ricci flow and VIB to deduce a curvature optimization approximation to form a tractable IB objective function. Extensive experiments on various datasets demonstrate the superior effectiveness and interpretability of CurvGIB.

Figures

Figures reproduced from arXiv: 2412.19993 by the authors.

Figure 1
Figure 1. Ricci flow, Ollivier-Ricci curvature and different [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Instances of the geometric perspective, comparison different paradigms for graph structure refinement. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The proposed CurvGIB principle and its overall framework. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Parameter sensity and robustness analysis on Cora and Citeseer. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Learning Stability Analysis on Different Datasets [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Distributions of Ricci curvature and visualizations of sampling sub-structures after learning on Cora. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Reference graph

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