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Train on small graphs, generate big ones: diffusion on graphons scales up

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2026-07-09 16:34 UTC pith:VSYUBGDA

load-bearing objection First diffusion model defined directly on graphon space; moment-matching theory is clean but the reverse-process score approximation is uncharacterized. the 4 major comments →

arxiv 2607.07232 v1 pith:VSYUBGDA submitted 2026-07-08 stat.ML cs.AIcs.LG

DiPhon: Diffusion on Graphons for Scalable Graph Generation

Sergio Rozada , Yiming Qin , Manuel Madeira , Pascal Frossard , Alejandro Ribeiro This is my paper
classification stat.ML cs.AIcs.LG
keywords graphprocessdiphondiffusiongenerationgraphsgraphoncontinuous
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

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.

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 (4)
  1. 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 (
  2. 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.
  3. 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.
  4. 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.
minor comments (7)
  1. 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.
  2. 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).
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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...').

Circularity Check

0 steps flagged

No circularity: moment-matching is a design choice, not a self-definitional reduction; the conditional score is derived from independent spectral theory.

full rationale

The paper's derivation chain is self-contained and does not exhibit circularity. The central theoretical result (Theorem 1) states that the renormalized discrete SDE (Eq. 6) with parameters (Eq. 7) matches the first two moments of the continuous graphon diffusion's spatial average (Eq. 5). This is a design choice (choosing parameters to match moments), not circular reasoning: the continuous process (Eq. 4) is defined independently via the Jacobi SDE, and the discrete process is a separate construction whose moments are then proven to align. The conditional score (Eq. 17) is derived from the known spectral eigendecomposition of the Jacobi diffusion (Appendix B.3), which is classical mathematics [28, 48, 54], not a self-citation. The denoiser is trained via a standard binary cross-entropy loss (Eq. 18) to predict the clean graph, and its output is plugged into the conditional score formula as a surrogate for the unobserved graphon entry. While the skeptic correctly notes that this plug-in substitution is inexact for nonlinear Jacobi dynamics (unlike Gaussian diffusion), this is a correctness/approximation concern, not circularity: the paper does not claim the marginal score equals the conditional score, nor does it define one in terms of the other. The experiments use external baselines (DiGress, DeFoG, GDSS, GruM) and standard graph families (SBM, PA, trees), providing independent validation. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 2 invented entities

The free parameters (κ, σ, µ, T) are standard diffusion hyperparameters, not fitted constants. The key ad-hoc axiom is the plug-in approximation of W_0 by the denoiser output, which lacks a formal error bound. The invented entities (DiPhon, renormalization) are mathematical constructions with falsifiable properties, not postulated physical objects.

free parameters (6)
  • κ (mean reversion rate) = 1.0 (in experiments)
    Controls drift toward stationary mean; set to 1 in SBM experiments (Appendix G.1).
  • σ (noise amplitude) = 0.005 (in experiments)
    Controls diffusion strength; set to 0.005 in SBM experiments. Must satisfy stability condition κ̃ > 0.
  • µ (stationary mean) = 0.45 (SBM), data-dependent
    Mean of Beta stationary law; set to (p+q)/2 for SBM experiments.
  • T (diffusion horizon) = 4.0
    Terminal time for forward/reverse integration.
  • M (Jacobi mode truncation) = not specified
    Truncation order for the score series in Eq. 17; used in Algorithm 2 but value not reported.
  • K (reverse steps) = not specified
    Number of Euler-Maruyama steps in reverse integration; affects discretization error.
axioms (5)
  • standard math Graphons are the canonical limit objects of dense graph sequences (Lovász–Szegedy theory)
    Invoked in Section 3.1 to justify graphon as the size-agnostic representation; standard result from [5, 6].
  • standard math Anderson's time-reversal theorem for diffusions
    Used in Section 4.1 to derive the reverse-time SDE (Eq. 15); classical result from [27].
  • standard math Jacobi diffusion has Beta stationary law and admits spectral expansion in Jacobi polynomials
    Used throughout Section 4 and Appendix B to derive the conditional score; standard from [48, 54].
  • domain assumption Observed graph A_0 is a Bernoulli sample of an underlying graphon
    Stated in Eq. (10); assumes the graph data is consistent with a graphon model, which may not hold for sparse graphs.
  • ad hoc to paper The denoiser output Φ_θ is a sufficient proxy for the unobserved graphon entry W_0^ij
    Section 4.2 replaces W_0^ij with the GNN prediction Â_0 in the score formula (17); the error from this substitution is not bounded.
invented entities (2)
  • DiPhon (the discretized graph-level Jacobi process) independent evidence
    purpose: Finite-dimensional approximation of the graphon diffusion that can operate on observed graphs
    The moment-matching properties (Theorem 1, Corollary 1) provide falsifiable mathematical guarantees that can be checked against the continuous process; empirical validation in Figure 2 confirms the moment predictions.
  • Renormalized parameters (σ̃, κ̃, μ̃_t) in Eq. (7) independent evidence
    purpose: Adjust the discrete Jacobi SDE so its moments match the spatially-averaged continuous process
    The renormalization is derived from the moment ODEs and verified empirically; the variance discrepancy (Eq. 9) is a closed-form falsifiable prediction.

pith-pipeline@v1.1.0-glm · 35491 in / 2949 out tokens · 467303 ms · 2026-07-09T16:34:11.805290+00:00 · methodology

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read the original abstract

Diffusion models represent a leading paradigm for graph generation, with notable impact in domains such as molecular design. Yet, scaling these models to large graphs remains an open problem. We approach this question in the dense-graph setting through the lens of graphons, the size-agnostic limit objects of dense graph sequences, to study how structural graph statistics behave across node-size scales. This perspective leads to DiPhon, a diffusion framework for size-scalable graph generation. Specifically, we formulate a continuous diffusion process on the graphon space via a Jacobi stochastic differential equation (SDE), and propose DiPhon, a discretized graph-level process that mimics these dynamics on finite graphs. We further derive the corresponding reverse-time process, which requires access to the marginal score. For the Jacobi process, this score interestingly admits a tractable form, which we estimate from data via graph denoising and plug into the reverse process to generate graph samples. We prove that DiPhon matches exactly the first moment of the marginal distributions induced by the continuous graphon process, and approximates the second moment up to a closed-form discrepancy. Thus, DiPhon inherits key size-agnostic statistical properties of the graphon dynamics, providing a principled route toward scalable graph generation. Empirically, we demonstrate this scalability by training on small graphs and generating progressively larger graphs at inference time, without retraining, while preserving their core topological properties.

Figures

Figures reproduced from arXiv: 2607.07232 by Alejandro Ribeiro, Manuel Madeira, Pascal Frossard, Sergio Rozada, Yiming Qin.

Figure 1
Figure 1. Figure 1: Schematic of DiPhon. The top row shows the original graphon diffusion process, the middle [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: SBM intra-community cell, σ = 0.005, T = 4, M = 103 . Top: N = 20. Bottom: N = 100. Empirical moments of (6) (solid) match the closed-form references (dashed/dotted) under both deterministic and Bernoulli initial conditions. renormalized discrete process coincide. We set κ = 1, σ = 0.005, T = 4, and simulate (6) with Euler–Maruyama using nsteps = 400 and M = 1000 Monte Carlo trajectories for different size… view at source ↗
Figure 3
Figure 3. Figure 3: Out-of-distribution size generalization. DiPhon maintains high accuracy across number of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SBM, intra-community cell, N = 10 [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: SBM, intra-community cell, N = 20 [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: SBM, intra-community cell, N = 50 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: SBM, intra-community cell, N = 100. Inter-community cell (W¯ ij 0 = 0.2) [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: SBM, inter-community cell, N = 10. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: SBM, inter-community cell, N = 20 [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: SBM, inter-community cell, N = 50 [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: SBM, inter-community cell, N = 100. G.2 Scalability experiments The scalability experiment evaluates whether DiPhon can generate graphs at node sizes outside the training distribution. To this end, we generate all datasets ourselves using standard graph generation tools such as networkx. For each graph family, we build a training set of 300 graphs, a validation set of 200 graphs, and multiple test sets of… view at source ↗
Figure 12
Figure 12. Figure 12: Uncurated SBM graph samples. Dataset N = 40 N = 60 N = 80 N = 100 N = 120 N = 140 N = 160 N = 180 N = 200 N = 220 N = 240 N = 260 N = 280 N = 300 DiGress DeFoG GDSS GruM DiPhon [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Uncurated PA graph samples. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Uncurated Tree graph samples. H Discussion and Limitations This work connects two lines of research that have so far been treated separately: diffusion-based generative models on graphs and the theory of graphons as the limit object of dense graph sequences. To the best of our knowledge, no previous diffusion model on graphs has placed the forward dynamics on graphon space. Prior works that use graphons i… view at source ↗

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