Title: Is Noise Conditioning Necessary? A Unified Theory of Unconditional Graph Diffusion Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph Diffusion Models

Unconditional Graph Diffusion

Noise Conditioning
Abstract: Explicit noise-level conditioning is widely regarded as essential for the effective operation of Graph Diffusion Models (GDMs). In this work, we challenge this assumption by investigating whether denoisers can implicitly infer noise levels directly from corrupted graph structures, potentially eliminating the need for explicit noise conditioning.  To this end, we develop a theoretical framework centered on Bernoulli edge-flip corruptions and extend it to encompass more complex scenarios involving coupled structure-attribute noise. Extensive empirical evaluations on both synthetic and real-world graph datasets, using models such as GDSS and DiGress, provide strong support for our theoretical findings. Notably, unconditional GDMs achieve performance comparable or superior to their conditioned counterparts, while also offering reductions in parameters (4-6%) and computation time (8-10%). Our results suggest that the high-dimensional nature of graph data itself often encodes sufficient information for the denoising process, opening avenues for simpler, more efficient GDM architectures.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce an extensive theoretical framework for quantifying and understanding the effects of noise conditioning on diffusion models applied for graph generation. The theory proposed within this paper establishes that not only is the noise level of the graph diffusion process inferable but that squared error scales and the reconstruction error is bounded under the paper's assumptions; indicating the ability of the diffusion process to encode information from graph structure. The authors further explore this effect with a coupled structure-feature noise model. Empirical analysis across synthetic scenarios and with DiGress, GDSS on QM9 and soc-Epinions1 provide consistent proof to the author's claims.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n---------\n1. The authors provide extensive theoretical and targeted empirical analysis on the effects of noise-conditioning with graph diffusion models indicating an intuitive relation with information stored within target graphs and the ability of the diffusion process to encode said information for models without explicit targeting to retain said information and high-downstream performance. The authors do this in a way that condenses already-dense information into an easily-parsed format.\n2. The impact of answering the question regarding the necessity of noise conditioning is significant in itself given that developments aided by this analysis are highly-likely to distinguish graph diffusion models from diffusion models within other data modalities, as evident within the author's exploration of structure-feature coupled noise modelling.\n3. The insights provided by the authors are well-supported by theory and experiments, aiding to the significance of this work. The author's also take care to indicate limitations of their theory and potential future applications in interesting contexts for reducing model parameters in expensive graph diffusion models.\n\nWeaknesses:\n----------\n1. As discussed in the author's limitation section, the assumptions within this paper are not unreasonable but are fairly-strong (i.e. power-law scaling/degree constraint in input graphs, Lipschitz denoisers). This can make it difficult for practitioner's to quantify how to design an effective graph diffusion model following intuitions derived from this paper. However, this does not detract from paper quality or significance. More concerning details are explored within Question 1 of this review.\n2. The evaluation of the coupled structure-feature graph generative model is comprehensive in it's relation to theoretical assumptions but seems to have limited practical applications. More details are provided within Questions 2 and 3 of this review.\n3. There are minor clarity issues within the main paper's Section 5 and 6 given the omission of details, but these are addressed with more extensive details provided within the paper's appendix.\n\nQUESTIONS:\n1. Why chose classification accuracy and node reconstruction error as the target for testing the coupled structure-feature diffusion model? From what I gathered in the reading, it appears that the Section 5 theory and subsequently-built latent process allow the coupled gaussian noise process to model the generation of features as well as structure. Within pre-existing graph diffusion models, the modelling of features and structure is often detailed as two separate downstream tasks; 'general' graph generation measured with MMD (Section 7.1 and Table 1 in [1]) and node regression measured with MAE (Section 5.2.1 and Table 3 in [2]) to indicate effective learning of node features. The node regression is not necessatily 1-to-1 since it typically involves some form of guidance in pre-existing graph generative models but the insights testing this mechanism as it relates to this paper feels especially pertinent given how pre-existing models rely on conditional-generation for effective regressions. I understand that strong classification performance can be foundation for developing meaningful graph generative models, but the seemingly-high reconstruction error makes me question the applicability of this framework and insights to directly generating graphs. This is the first of two defining limiting factors for this current review score, particularly since it's difficult to quantify the practical benefit propsed by this paper without further clarification as to why there is or can not be a 1-to-1 application of metrics commonly used in other graph generative architectures.\n\n2. What practical considerations would have to be made in order to derive more formal proofs when considering degree or power-law scaling constraint for graph data, as indicated within Appendix A and F. From there, it seems possible to fully-quantify whether the issue which occurs within the warmed-up version of DiGress when not conditioned on the soc-Epinions1 Dataset is more closely-related to a reliance on explicit time-conditioning or another potential shortcoming within how DiGress can effectively model graph generation. This consideration is not necessarily a defining factor for the score within this review, but seems important to consider in order to reduce the potential for confirmation bias. This can be addressed through clarifications as they relate to this paper's conceptual outlines or theory.\n\n3. Additionally, although the evaluation is comprehensive in how well it relates to the theoretical components of this paper; two datasets seems limited in terms of determining whether increased graph signals does indeed contribute to the idea that score-based diffusion models like GDSS are more resilient to the removal of explicit time-free conditioning versus DiGress, even when altering the size of maximum subgraph node sizes from N=50 to N=200. Is it not viable to repeat or extend the tests conducted within Table 2 and 7 to include synthetic Barabasi-Albert graph samples with varying weights applied to the preferential-attachment mechanism? From there, it seems possible to form and test for a more continuous trend between the relation of graph signals and the resilience of scoring-diffusion models to the removal of time-conditioning. This question can be address through clarification as to why these tests are excluded, but direct results would be certain to eliminate any potential bias. This is the second of the two defining factors for the score within this review, any relevant clarifications or additional tests are certain to raise the score within this review.\n\n[1] Vignac, Clement, Igor Krawczuk, Antoine Siraudin, Bohan Wang, Volkan Cevher, and Pascal Frossard. \"Digress: Discrete denoising diffusion for graph generation.\" arXiv preprint arXiv:2209.14734 (2022).  \n[2] Cai, Zhou, Xiyuan Wang, and Muhan Zhang. \"Latent graph diffusion: A unified framework for generation and prediction on graphs.\" arXiv e-prints (2024): arXiv-2402.\n\nLIMITATIONS:\nYes, the authors explore the limitations of their work directly within Section J of the Appendix. Although I agree with the assertion that including more detailed proofs for error bounds on Poisson, Beta, and Multinomial noise will produce a more comprehensive paper; the current theoretical and empirical results still indicate promise which is insightful and complete enough for the further development/improvement of diffusion models for graph generation.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper challenges the prevailing assumption that explicit noise-level (or timestep) conditioning is essential for Graph Diffusion Models (GDMs). The authors develop a unified theoretical framework to argue that for discrete graph data, the denoising model can implicitly infer the noise level directly from the corrupted graph structure. The core of their theory is built on three results: Edge-Flip Posterior Concentration (EFPC), which shows the noise level is inferable with posterior variance shrinking as $O\\left(M^{-1}\\right)$ with the number of potential edges M; Edge-Target Deviation Bound (ETDB), which bounds the single-step error from omitting conditioning by $O\\left(M^{-1}\\right)$; and Multi-Step Denoising Error Propagation (MDEP), which proves these small errors accumulate linearly, leading to a manageable total error of $O\\left(T M^{-1}\\right)$.\n\nThis framework is extended to handle more complex scenarios, including coupled structure-attribute noise. To validate their theory, the authors conduct extensive experiments. First, on synthetic data, they show an excellent match between their theoretical scaling laws and empirical measurements. Second, on real-world benchmarks for molecular generation (QM9) and social network generation (soc-Epinions1), they compare standard \"t-aware\" variants of GDSS and DiGress with \"t-free\" counterparts. The results demonstrate that unconditional GDMs can achieve comparable or even superior performance, particularly on larger graphs, while also being more efficient by reducing parameter counts (4-6%) and computation time (8-10%).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper tackles a foundational assumption in a popular and powerful class of generative models. Moving away from explicit noise conditioning could simplify GDM architectures, improve efficiency, and is a significant conceptual contribution. While blind denoising has been explored in continuous domains such as images, this work provides a novel and non-trivial extension to the discrete, structured domain of graphs, supported by a new theoretical framework.\n* The paper's claims are backed by a rigorous and comprehensive theoretical framework (EFPC, ETDB, MDEP). The proofs are detailed extensively in the appendices and the theory is thoughtfully extended to more complex cases like coupled feature noise and scale-free graphs, demonstrating a deep engagement with the problem.\n* The empirical validation is well-executed.\n   - The direct validation of the theoretical scaling laws on synthetic data is a major strength, with the tight agreement between theory and experiment (e.g.,$R^2 \\geq 0.995$) providing strong evidence for the framework's correctness.\n   - The evaluation on real-world datasets using two distinct, state-of-the-art GDM architectures (GDSS and DiGress) grounds the work in practical relevance.\n   - The analysis is nuanced, exploring different training strategies (from scratch vs. warm-starting) and varying graph sizes, which provides valuable insights into the practical applicability of the approach.\n* Clarity: The paper is very well-written and structured. The central research question is clearly motivated and stated , and the logical progression of the theoretical argument is easy to follow. Figures and tables are clear, informative, and effectively support the paper's claims.\n\nWeaknesses:\n* The theoretical results hinge on key assumptions, notably that the learned denoiser is Lipschitz with a constant close to 1 and that the model can achieve a single-step prediction error of $O\\left(M^{-1}\\right)$. While the authors acknowledge this in their limitations section, the practical verification of these assumptions (especially the Lipschitz constant of the trained networks) is not fully explored, which makes the link between the theory and the empirical results slightly less direct.\n* The success of the unconditional approach is not universal across all experiments, which slightly tempers the main claim that conditioning is \"unnecessary.\" For example, the warm-started DiGress model shows a catastrophic failure in validity on the soc-Epinions1 dataset. Furthermore, the t-free model trained from scratch, while promising, often lags behind the t-aware or warm-started variants in fine-grained structural metrics (MMDs). The conclusion seems to be more that conditioning is \"not always necessary\" rather than completely redundant.\n* A potential weakness of the paper is that the empirical validation, while thorough for the models tested, does not cover the full taxonomy of modern Graph Diffusion Models (GDMs). The experiments focus on GDSS , a continuous-state/continuous-time model, and DiGress, a discrete-state/discrete-time model. This leaves the framework's applicability to other significant GDM classes empirically unverified. For instance, the authors do not test their framework on discrete-state, continuous-time models, such as the recently proposed DISCO [1] and continuous-state, discrete-time models, a category that includes influential works like EDP-GNN [2]. While the authors briefly sketch a theoretical extension to Poisson processes (which underpins DISCO), providing empirical results for these other model classes would be crucial to fully substantiate the central claim of developing a \"unified theory\" and demonstrating that noise conditioning is broadly unnecessary.\n\n[1] Xu, Zhe, et al. \"Discrete-state Continuous-time Diffusion for Graph Generation.\" arXiv preprint arXiv:2405.11416 (2024).\n\n[2] Niu, Chenhao, et al. \"Permutation invariant graph generation via score-based generative modeling.\" International conference on artificial intelligence and statistics. PMLR, 2020.\n\nQUESTIONS:\n1.\tThe paper notes that the t-free(warm) DiGress model's validity collapses on soc-Epinions1, attributing this to the transformer architecture's reliance on time embeddings. However, many state-of-the-art GDMs, including DiGress itself and others like DISCO-GT [1], leverage transformers for their strong performance. Does this failure imply that the proposed implicit noise inference is incompatible with top-performing GDM architectures? Could the authors elaborate on why this failure occurs? For instance, does the attention mechanism over-fit to the time embedding, causing an unrecoverable collapse when it is removed? Understanding this specific failure mode is crucial for assessing the practical scope of the unconditional approach.\n2.\tThe theory posits that error scales inversely with M, the number of potential edges. The experiments on soc-Epinions1 show that increasing subgraph size from Nmax=50 to Nmax=200 drastically improved validity for all GDSS models. Is this improvement purely a function of the larger M, or is there a more subtle interplay with the \"richness\" or \"complexity\" of the graph structure that isn't captured by M alone? For instance, how would the theory and models perform on a very large but extremely sparse graph, where M is large but the number of actual edges (and thus structural information) is low?\n3.\tThe paper provides a comprehensive and rigorous quantitative evaluation using statistical metrics like MMD and chemical property distributions. However, the evaluation currently lacks a qualitative component. To provide a more intuitive and complete comparison, have the authors considered including visualizations of representative graph samples? For instance, showing a few valid molecules generated by the t-aware and t-free variants of GDSS/DiGress on QM9, or visualizing typical subgraphs from the soc-Epinions1 experiments, could be very insightful. \n4.\tIn Table 2 (QM9 Dataset), the t-aware DiGress variant is reported with a slightly lower per-epoch computation time (47.82s) than the t-free (48.16s) and t-free(warm) (48.23s) variants. This is counter-intuitive, as the t-free models have fewer parameters  and should theoretically be faster. Could the authors clarify the reason for this discrepancy? Is it due to experimental variance, or is there a specific implementation detail causing this minor overhead?\n5.\tThe real-world experiments are conducted on the QM9 dataset and soc-Epinions1 dataset. To better assess the practical advantages of the t-free approach, have the authors considered evaluating their framework on benchmarks with larger and more complex graphs or molecules, such as those found in the GuacaMol benchmark [3]? Such experiments would provide stronger evidence for the method's scalability and its applicability to more challenging generation tasks.\n\n[3] Brown, Nathan, et al. \"GuacaMol: benchmarking models for de novo molecular design.\" Journal of chemical information and modeling 59.3 (2019): 1096-1108.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis works investigates the necessity of explicit noise conditioning in graph diffusion models (GDMs). While several recent studies have addressed this issue for continuous data, graph data inherently involves discrete types of noise, making the direct application of previous studies nontrivial.\nThe authors first highlight a theoretical analysis indicating that a noisy graph implicitly encodes the information about the noise itself. Based on this, they further explain that omitting explicit noise conditioning leads to small errors that accumulate linearly during the generation process.\nConsequently, the authors present empirical evidence that supports the theoretical results in both synthetic and real-world data setting. In sum, the authors conclude that explicit noise conditioning is often unnecessary for graph generation.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n1. This work studies an important question; expanding the investigation of explicit noise conditioning to discrete, structured data.\n2. Theoretical framework are well-structured. Which starts from the posterior concentration to the final error propagation, and expanding it to the coupled structure-feature noise model that tries to model the real world.\n3. Empirical results are provided to support the theoretical investigation.\n\n### Weaknesses\n1. There are several points in the current version of this draft where it seems overstated. For example, 'unified theory' in the title and 'extensive' in the abstract. Although the authors provide a scetch how the current theoretical results extends to other types of noise corruption, I don't see as a 'unified' framework in this current stage. \n2. While I agree that investigating the necessity of explicit noise conditioning in GDMs is an important topic to explore, the writing seems to lack clear motivation. Especially in the Introduction line 24-29 is written in result-oriented way and makes a logical leap before introducing the central question in line 30. Furthermore, this work might be seen as a special case study of Ref. [10] (the flow of the proofs are inevitably similar). The authors should more focus on the different nature of the noise inherent in the common diffusion models and graph diffusion models.\n3. There are some points that the authors claimed which the empirical justifications should be more strenghtened (refer to Questions).\n\nQUESTIONS:\nBelow are some points that disturbed me from understanding the paper. I would highly appreciate if the authors could address the following questions.\n\n### Minor questions\n\n1. How does the Ref. [13] relate to the statement in line 23-24?\n2. It would be nice if the authors provide an appropriate reference corresponding to the statement of line 51-52.\n3. In the part describing the forward process using the Bernoulli edge-flipping model (line 81-83), shouldn't be $\\beta_t \\in [0, \\frac{1}{2}]$ to make $\\tilde{A}_T$ approach to a random graph (assuming the authors meant ER network in this context)? Please correct me if I'm wrong.\n4. In line 266 or 274, the notation $|E|$ is used as a potential edge, while $M$ was primarily used in this context beforehand. Are these two used differently or is this a typo?\n5. Since the authors framework is based on Bernoulli edge-flip, benchmark experiments with GraphGUIDE [15] would be a natural model to explore before expanding it into two architectures; DiGress [7] and GDSS [18].\n6. Minor suggestion. It would have been more readable if the core part of the experiment setting was described in the main body, or at least if the reference to the corresponding part of the appendix was made. For example, which type of network was used in which setting. I had to go through the appendix to find that SBM was used in uncoupled, and ER was used in coupled setting.\n\n\n### Major questions\n1. Also related to $\\beta_t$, has the authors experimented on varying noise schedule in synthetic setting? Showing the robustness with respect to $\\beta_t$ would more strengthen the theoretical framework to the real-world practices.\n2. In Section 4.1, the authors explicitly provide the EFPC for scale-free network. Then can the authors also provide the empirical results for scale-free case? If this was provided, it would highly support the authors claims.\n3. In the synthetic experiment setup, is there a specific reason to only use SBMs in the uncoupled case and ER network in the coupled case? Is the empirical results robust across graph structures (i.e. for the uncoupled model, is the results also consistent for example, ER and scale-free networks)?\n4. I understand that the authors used shared latent random vectors to model the coupled structure-feature dynamics of real networks. Then I think it would be (slightly) more reasonable to experiment with scale-free networks than ER networks in their synthetic setup of the coupled model, if they were to take real-world into account. According to line 1113-1115 in the appendix, it is said that ER network is used in this setting.\n5. In Section 5.1, while the authors model the noisy adjacency matrix as continuous, it seems that in the experiments, Bernoulli flips were used for adding the noise (line 1125 in the appendix). Can the authors provide a detailed discussion about this discrepancy (for example, possible effects to empirical results)?\n6. Regarding the results of Section 6.3, the authors highlight that stronger coupling \"improves\" both reconstruction error and task performance. What I do not understand about this setting is that the coupling coefficient $\\gamma$ is used as a factor to model the real-world networks, not as a control variable in the authors framework. Although the result that shows the dependence of error/performance on $\\gamma$ despite that in theoretical result it is not provided, is interesting, I think the experiment is slightly misconducted in this perspective. Also, I think that experiments in this section should first support the fact that JPC or JMEP stays valid irrespective of $\\gamma$ before claiming the statement made in line 296-298.\n\nLIMITATIONS:\nYes, however the limitations are currently discussed in the appendices. It would be preferable to move the relevant content into the main body of the paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper investigates whether explicit noise-level conditioning is necessary in graph diffusion models (GDMs). Through new theoretical analysis and extensive experiments, the authors show that unconditional GDMs can match or even surpass conditioned models in both accuracy and efficiency (4–6% fewer parameters, 8–10% faster runtime) without explicit noise input. This finding challenges a key assumption in diffusion model design and suggests simpler, more effective approaches for graph generative modeling.

STRENGTHS AND WEAKNESSES:
## Strengths
This paper provides a novel theoretical framework demonstrating that explicit noise conditioning is not necessary for large-scale GDMs, as the models can infer noise levels implicitly. Backed by solid mathematical proofs and extensive experiments on both synthetic and real datasets, the work not only advances our understanding but also offers practical benefits without sacrificing performance.

## Weaknesses
The paper mainly addresses theoretical derivations and preliminary experiments. Its performance and robustness in complex or industrial-scale applications remain to be thoroughly investigated.

QUESTIONS:
What are the limitations of unconditional models in tasks that require fine-grained or conditional generation, such as node-specific or class-conditional graph generation?

LIMITATIONS:
See Weaknesses.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03