{"id":"c308a21d-cf8e-47da-9541-3336058ca4d5","arxiv_id":"2607.07232","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":6,"one_line_summary":"A Jacobi diffusion on graphon space is discretized into a graph-level generative process that matches the continuous process's first moment exactly and second moment up to a closed-form gap, enabling out-of-scale graph generation.","lead":"The paper builds a graph diffusion model on graphons—mathematical limit objects of large graphs—so it can train on small graphs and generate larger ones without retraining. A smart generalist might read it because size-transferable generative models could reduce compute costs in molecular design and network science.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The reverse process substitutes a plug-in conditional score for the required marginal score; this substitution is inexact for Jacobi dynamics (unlike Gaussian diffusion) and the error is unbounded, potentially undermining the generative claim.","rationale":"The reader identifies the moment-matching gap (only first two moments, no full distributional bound) as the weakest assumption. This is valid but somewhat indirect: the forward process moment-matching is proven cleanly, and the concern that higher moments matter is speculative without a specific failure mechanism. I identify a more concrete version of the same theory-practice gap: the reverse process score approximation. The marginal score required by Eq. 15 is replaced by a plug-in conditional score (Eq. 17 with Â_0), and this substitution is provably inexact for nonlinear score functions—exactly the case for Jacobi diffusion. This is the specific mechanism through which 'moment-matching suffices for generative quality' could fail. The reader's concern and mine both point to the same underlying issue (forward-process theory doesn't guarantee reverse-process correctness), but through different and complementary angles. The reader focuses on the forward process; I focus on the reverse process. Both are needed for the central claim to hold. I keep the verdict at CONDITIONAL because the forward-process theory is sound and the empirical results are promising, but the unbounded score approximation error and the restriction to small-σ experiments leave the scalability claim insufficiently validated. The paper would need either (a) a bound on the plug-in score error, or (b) empirical validation across a range of σ values and on standard graph generation benchmarks, to move toward ACCEPT.","tokens_in":32673,"tokens_out":4559,"duration_ms":246515,"concrete_test":"For a two-community SBM graphon where W_0^{ij} ∈ {p, q} with known probabilities, compute the exact marginal score numerically via ∇_W log[π_t(W̃_t|p)·P(p) + π_t(W̃_t|q)·P(q)] (truncating the spectral expansion at finite M) and compare it to the plug-in conditional score ∇_W log π_t(W̃_t|Â_0) across a range of σ values (e.g., σ ∈ {0.005, 0.05, 0.2, 0.5}) and diffusion times t. If the L² distance between marginal and plug-in scores grows substantially with σ, the method's scalability claim is restricted to the small-σ regime and the generative quality may degrade for larger noise.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reverse-time SDE (Eq. 15) requires the marginal score ∇_W log p̃_t(W̃_t^{ij}). The paper instead computes the *conditional* score ∇_W log p̃_t(W̃_t^{ij} | W_0^{ij}) from Eq. 17, with W_0^{ij} replaced by the denoiser's point estimate Â_0^{ij} (Section 4.2). This plug-in is exact for Gaussian diffusion because the conditional score is linear in x_0, so E[score(x|x_0)|x] = score(x|E[x_0|x]). For Jacobi diffusion, the conditional score (Eq. 17) is a nonlinear function of W_0 through the Jacobi polynomials ϕ_n(W_0) appearing in both numerator and denominator. Therefore ∇_W log p̃_t(W̃_t) ≠ ∇_W log p̃_t(W̃_t | E[W_0|W̃_t]) in general, and the substitution introduces an error that the paper neither bounds nor acknowledges. This matters because the generative quality—the actual claim of 'size-scalable graph generation'—depends on the reverse process being correct, not just the forward moment matching. The experiments use σ=0.005 (Section 5.1), a regime where the diffusion is nearly deterministic and the conditional score is approximately linear in W_0, so the plug-in may work well there but fail for larger noise. The concern is not that the approximation is unreasonable, but that its regime of validity is uncharacterized, so the scalability claim rests on an untested assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper introduces DiPhon, a diffusion-based graph generative model designed for size-scalable generation. The core idea is to define a forward diffusion process on graphon space—the limit object of dense graph sequences—using a Jacobi SDE whose state is naturally bounded in [0,1], making it compatible with edge-probability semantics. The authors then discretize this process to obtain a tractable graph-level SDE, derive the reverse-time process, and propose a training scheme based on graph denoising. The main theoretical contributions are: (1) Theorem 1 and Corollary 1, which show that the discretized process matches the first moment of the continuous graphon process exactly and the second moment up to a closed-form discrepancy; and (2) Theorem 2, which establishes that the stationary distribution of the sampled graphs converges in homomorphism density to an Erdős–Rényi graphon. Empirically, the authors demonstrate that DiPhon can be trained on small graphs and generate larger graphs at inference time while preserving structural properties, outperforming discrete and Gaussian diffusion baselines on SBM, PA, and tree datasets.","tokens_in":33244,"tokens_out":1911,"duration_ms":429538,"significance":"The paper addresses a genuinely important and underexplored problem: transferring generative models across graph sizes without retraining. The graphon perspective is a natural and principled bridge for this, and the use of Jacobi diffusion to maintain bounded edge probabilities is a well-motivated alternative to Gaussian SDEs. The moment-matching theorems are clean, the closed-form score via Jacobi polynomial expansions is a notable technical contribution, and the Erdős–Rényi limit theorem provides a useful asymptotic sanity check. The empirical results on out-of-distribution size generalization are promising, particularly the robustness on sparse tree graphs outside the formal dense-graphon regime. The framework is specific and falsifiable, with clearly identified parameters and assumptions.","major_comments":[{"comment":"Section 4.1–4.2, Eqs. (15)–(17): The reverse-time SDE (Eq. 15) requires the marginal score ∇_W log p̃_t(W̃_t^{ij}), but the paper substitutes the conditional score ∇_W log p̃_t(W̃_t^{ij} | W_0^{ij}) from Eq. (17), with W_0^{ij} replaced by the denoiser's point estimate Â_0^{ij}. For Gaussian diffusion, the conditional score is linear in x_0, so substituting E[x_0|x_t] is exact. For the Jacobi process, the conditional score (Eq. 17) is a nonlinear function of W_0 through the Jacobi polynomials ϕ_n(W_0) in both numerator and denominator. Thus ∇_W log p̃_t(W̃_t) ≠ ∇_W log p̃_t(W̃_t | E[W_0|W̃_t]) in general. The paper does not bound or characterize the error introduced by this plug-in approximation. This is load-bearing for the generative claim, since the scalability of generation depends on the reverse process being correct, not just the forward moment matching. The authors should either (","section":null},{"comment":"Section 5.1, Figure 2: The forward moment-matching experiments use σ=0.005, which is an extremely small noise amplitude. In this regime, the Jacobi diffusion is nearly deterministic and the conditional score is approximately linear in W_0, so the plug-in approximation discussed above may work well by coincidence rather than by the mechanism the paper intends. The scalability experiments in Section 5.2 also appear to use constant-parameter settings, but the specific σ value is not reported. The authors should clarify what σ values are used in the generative experiments and whether the moment-matching and score-approximation guarantees remain meaningful in those regimes.","section":null},{"comment":"Section 6, Theorem 2: The Erdős–Rényi limit theorem (Eq. 14) shows that the forward process converges to a trivial stationary distribution (i.i.d. Bernoulli with mean µ). While this is a valid asymptotic check, it means the forward process destroys all graph structure at stationarity. The paper does not discuss how this relates to the generative quality of the reverse process, which must recover structure from this uninformative starting point. The connection between the trivial stationary law and the practical difficulty of the reverse denoising task should be addressed, especially given that the score approximation error (Major Comment 1) may compound when the reverse process must traverse from an unstructured state.","section":null},{"comment":"Section 5.2, Figure 3: The evaluation metrics (forest accuracy for trees, statistical validity for SBMs, power-law fit for PAs) measure whether generated graphs satisfy the defining property of each family, but do not measure distributional fidelity (e.g., whether the generated graphs match the training distribution beyond the coarse property). A method could achieve high accuracy by generating trivially valid but distributionally wrong graphs (e.g., a forest of isolated edges). The authors should include at least one distributional metric (e.g., degree distribution distance, clustering coefficient comparison) to strengthen the scalability claim.","section":null}],"minor_comments":[{"comment":"Section 3.3, Eq. (7): The renormalization formula for κ̃ involves the term σ²(N²−1)/2. The stability condition κ̃ > 0 is mentioned in Appendix C.2 but not in the main text. This condition constrains the admissible (κ, σ, N) combinations and should be stated in Section 3.3.","section":null},{"comment":"Section 4.2, Eq. (18): The denoising loss uses A_0^{ij} as the target, but the text in Section 4.2 argues that A_0 is an unbiased proxy for the unobserved graphon entry W_0^{ij}. The loss minimizes E[||Φ_θ − A_0||²] (in cross-entropy), which recovers E[A_0|W̃_t] = P(W_0=1|W̃_t), not W_0 itself. The paper should clarify whether the denoiser is intended to predict the binary edge or the continuous graphon entry, as this affects the interpretation of the plug-in in Eq. (17).","section":null},{"comment":"Appendix E, Algorithm 2, line 9: The reverse-time integration includes a clipping step ('Clip W̃_{t−Δt} to [0,1]'). Since the Jacobi diffusion is bounded by construction, clipping should not be necessary if the SDE is integrated correctly. The presence of clipping suggests numerical instability in the Euler–Maruyama scheme. The authors should discuss whether this is a practical concern and whether it affects the theoretical guarantees.","section":null},{"comment":"Section 5.2: The specific hyperparameters used for DiPhon in the scalability experiments (κ, σ, µ, T, K, M) are not reported in the main text. These should be included, at least in summary form, to allow reproducibility.","section":null},{"comment":"Figures 12–14 (Appendix G.2): The visualization grids are useful but the node sizes are not labeled on the axes, making it difficult to compare across methods at specific sizes. Adding size labels would improve clarity.","section":null},{"comment":"Section 2: The related work mentions that graphons have been used with graphette priors for flow matching [18], but does not discuss the relationship between DiPhon's graphon-space diffusion and that approach in sufficient detail. A sentence clarifying why placing the diffusion on graphon space (DiPhon) is fundamentally different from using graphons as a prior (Flowette) would help.","section":null},{"comment":"Typo in Section 3.3: 'maatches' should be 'matches' (in the sentence beginning 'we seek to design the discrete process such that the probability distribution of W̃_t^{ij} maatches...').","section":null}],"recommendation":"major_revision","confidential_remarks":"The core theoretical contribution (moment matching) is sound and the graphon perspective is well-motivated. However, the gap between the forward-process guarantees and the reverse-process approximation is the central weakness: the plug-in conditional score for the marginal score is uncharacterized, and the experiments use a very small σ that may mask the issue. The paper would be substantially strengthened by either bounding the approximation error or demonstrating empirically that the method works across a range of σ values. The sparse-graph results (trees) are interesting but the theory does not cover them, so the claims should be scoped accordingly. I would encourage the authors to address these points rather than withdraw, as the framework has clear merit."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The paper you should know about: DiPhon places a Jacobi diffusion directly on graphon space and discretizes it to get a graph-level generative process that can train on small graphs and sample larger ones without retraining. That is a genuinely new framing—nobody has put the forward dynamics on graphons before, and the bounded [0,1] range of the Jacobi SDE is the right tool for the job where Gaussian SDEs leak outside the probability range and discrete diffusions have no continuous spatial limit to connect to a graphon at all. The moment-matching theorem (Theorem 1, Corollary 1) is clean: the renormalized discrete process matches the continuous cell-average process exactly in mean and up to a closed-form, vanishing-in-time discrepancy in variance. The Erdős–Rényi homomorphism-density limit (Theorem 2) is a nice sanity check on the stationary regime. The conditional score formula via Jacobi polynomial eigenexpansion (Eq. 17) is a real technical contribution—tractable and closed-form. The empirical size-transfer results on SBMs, PA graphs, and trees are promising, especially the tree results outside the formal dense-graph regime. Now the soft spots. The stress-test concern about the plug-in score is the real one. The reverse SDE needs the marginal score, but the paper computes the conditional score with the denoiser's point estimate of W_0 substituted in. For Gaussian diffusion this is exact because the conditional score is linear in x_0. For Jacobi it is not—the Jacobi polynomials make it nonlinear in W_0, so the plug-in introduces an error the paper neither bounds nor acknowledges. The experiments use σ=0.005, a near-deterministic regime where the approximation is probably fine, but the regime of validity is uncharacterized. This is the load-bearing gap: the scalability claim depends on the reverse process being correct, not just the forward moment matching. The moment-matching equivalence itself is only first-two-moments—no full distributional distance bound—and the evaluation is synthetic-only with structural validity metrics rather than standard benchmarks. These are acknowledged in Section 6 but not resolved. Overall: the conceptual contribution is real and the theory is honest about its limits. A serious referee should push hard on the score approximation error and ask for either a bound or empirical evidence across noise regimes. Worth a careful review.","headline":"First diffusion model defined directly on graphon space; moment-matching theory is clean but the reverse-process score approximation is uncharacterized.","tokens_in":33765,"tokens_out":565,"would_cite":true,"duration_ms":176844,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Train on small graphs, generate big ones: diffusion on graphons scales up","keywords":[],"falsifier":"A graph family where the first two moments of the edge-probability distribution match between the continuous and discrete processes, but the generated graphs systematically fail to preserve a structural property that depends on third or higher-order moments, such as clustering coefficient or motif counts beyond edges.","tokens_in":32942,"feed_emoji":"🔬","tokens_out":1197,"duration_ms":160993,"temperature":0.7,"pith_summary":"Graph diffusion models generate graphs by progressively adding noise to edges and then learning to reverse that process. But the noise processes used today either operate on discrete edge states with no notion of a continuous limit, or use Gaussian noise that sends edge probabilities outside the [0,1] range. Neither connects naturally to graphons, the mathematical limit objects that describe what a sequence of dense graphs converges to as the number of nodes grows. This paper defines a diffusion process directly on graphon space using a Jacobi stochastic differential equation, a type of noise process that stays bounded in [0,1] by construction and has a Beta distribution as its stationary law. The key challenge is that the continuous graphon process involves space-time white noise, which cannot be evaluated pointwise. The authors get around this by discretizing the graphon into cells, assigning an independent Jacobi diffusion to each cell, and carefully renormalizing the diffusion parameters so that the discrete process matches the continuous one. They prove that this discretized process, called DiPhon, matches the first moment of the continuous graphon process exactly and matches the second moment up to a closed-form discrepancy that vanishes in the stationary regime. Because the process is defined on graphons, which are size-agnostic, a model trained on graphs of one size can in principle generate graphs of a different size without retraining. The reverse-time process requires the marginal score, which for the Jacobi diffusion admits a closed-form expression via an eigenexpansion in Jacobi polynomials, reducing training to a graph denoising problem. Experiments on stochastic block models, preferential attachment graphs, and trees show that DiPhon trained on small graphs can generate larger graphs while preserving structural properties, where competing discrete and Gaussian diffusion methods collapse at larger sizes.","feed_headline":"Train on small graphs, generate big ones: diffusion on graphons scales up","feed_subtitle":"A Jacobi diffusion defined on graphon space lets a model trained on 40-node graphs generate structurally valid 300-node graphs without re","key_machinery":"The key machinery is the renormalization of the Jacobi diffusion parameters that makes the discretize-then-diffuse construction match the diffuse-then-discretize construction. The continuous graphon process uses parameters (kappa, sigma, mu). The discrete cell-level process uses renormalized parameters where the noise is scaled by N^2, the mean-reversion rate is adjusted by a term involving (N^2 - 1), and the target mean is made time-varying to preserve the mean trajectory exactly. This renormalization ensures that the discrete process matches the first moment of the continuous process exactly and the second moment up to a discrepancy proportional to the in-cell variance of the initial graph","core_discovery":"The central object is a Jacobi diffusion defined on graphon space. A graphon is a symmetric function from the unit square to [0,1] that serves as the limit of a sequence of dense graphs of growing size, with the function value at a point giving the probability of an edge between two nodes labeled by that point. The Jacobi diffusion is a stochastic process on [0,1] whose diffusion coefficient vanishes at the boundaries 0 and 1, keeping the process confined to the unit interval, and whose stationary distribution is a Beta distribution. By placing this diffusion on graphon space and then discretizing, the authors obtain a finite-dimensional process on graph cells that provably matches the first","pith_inferences":["The moment-matching guarantee is established only for the first two moments. If generation quality depends on higher-order structure, two distributions sharing mean and variance could still produce structurally different graphs, and the size-transfer property might degrade for graph families where higher-order motifs dominate.","The theoretical framework is developed for dense graphs, but the experiments include sparse graph families like trees. The empirical success on sparse graphs suggests the size-transfer mechanism may be more broadly applicable than the theory formally justifies, but it is unclear why the mechanism works outside the dense regime.","The training overhead of simulating forward trajectories, since the renormalized Jacobi transition lacks a tractable closed-form density for direct sampling, could limit practical adoption relative to methods with tractable transitions."],"forward_implications":["If the moment-matching guarantee is sufficient for generation quality, then graph diffusion models can be trained on small graphs and deployed on large graphs, reducing the computational cost of training on large-graph datasets.","The Jacobi diffusion on graphon space provides a principled reference distribution for graph generation, analogous to the Gaussian reference in image diffusion, since the stationary regime converges to the Erdos-Renyi graphon in homomorphism density.","The framework could extend to node features, categorical edges, and weighted graphs by applying bounded Jacobi-type diffusions to those variables, though the theoretical guarantees would need to be re-derived.","The approach could inform graph neural network transferability, since the forward process produces graph-valued samples at every time step that could serve as data augmentation across graph sizes."],"fun_headline_variants":["Graphon diffusion: train small, generate large without retraining","Jacobi diffusion on graphons for scalable graph generation","Train on 40-node graphs, generate 300-node ones via graphon diffusion","Size-agnostic graph generation through Jacobi diffusion on graphons","Graphon-space diffusion preserves topology across graph scales"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The equivalence between the continuous graphon diffusion and the discrete process is proven only at the level of the first two moments. The full distance between the marginal distributions is not bounded, so if generation quality depends on higher-order structure, the size-transfer property may not hold.","fun_headline_variants_meta":{"raw":{"variants":["Graphon diffusion: train small, generate large without retraining","Jacobi diffusion on graphons for scalable graph generation","Train on 40-node graphs, generate 300-node ones via graphon diffusion","Size-agnostic graph generation through Jacobi diffusion on graphons","Graphon-space diffusion preserves topology across graph scales","Diffuse on graphons, generate graphs at any scale from small training"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":1516,"prompt_tokens":596,"completion_tokens":920,"prompt_tokens_details":null},"tokens_in":596,"tokens_out":920,"duration_ms":28489,"temperature":1.0,"reasoning_tokens":866,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T16:34:11.805290+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A graph family where the first two moments of the edge-probability distribution match between the continuous and discrete processes, but the generated graphs systematically fail to preserve a structural property that depends on third or higher-order moments, such as clustering coefficient or motif counts beyond edges.","supporting_citations":[],"review_version":1}