{"id":"b5d1b3dc-f0b2-44ea-bc32-0169b5f6c850","arxiv_id":"2505.19685","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"GGDiff casts conditional graph diffusion as stochastic optimal control and uses zero-order estimators to guide pre-trained models under differentiable and non-differentiable rewards.","lead":"This paper presents GGDiff, a method that steers pre-trained graph diffusion models toward user-defined rewards by treating guidance as a control problem, including a gradient-free version for non-differentiable rewards. It is a candidate generic plug-and-play conditioner for graph generation in drug design, fairness, and link prediction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"All GGDiff guidance rules inherit the Dirac-delta approximation in Eq. (5); if the one-step denoiser is inaccurate at coarse noise levels, the guidance direction is misaligned, and the paper provides no ablation or error analysis testing this failure mode.","rationale":"I considered two candidate weaknesses. First, the zero-order estimators are not truly derived from Eq. (4) and are acknowledged by the authors as surrogate directions; this weakens the 'unification' claim but is largely a framing issue. Second, the Dirac-delta replacement in Eq. (5) is the mathematical foundation for every variant, including the zero-order ones, because all of them evaluate the reward on \\hat G0(\\hat Gt). If that denoiser is inaccurate in the early reverse process, the guidance direction is misaligned for every variant, directly threatening the paper's central empirical claim of 'superior alignment with target rewards while maintaining diversity and fidelity.' This is the most load-bearing assumption: it is shared across GGDiff-G, GGDiff-C, and GGDiff-Z, and it is precisely the part of the derivation where the paper has no supporting analysis. The proposed ablation isolates the failure mode, and the error curve identifies the mechanism. The reader's weakest_assumption already points to this same approximation, so I agree with it. I do not see a reason to move the CONDITIONAL verdict: the concern is testable, the paper's current evidence does not rule it out, and the requested experiment would settle it.","tokens_in":17771,"tokens_out":10174,"duration_ms":109005,"concrete_test":"On the community-small max-degree and triangle-count tasks, run GGDiff-C (and GGDiff-G) in an ablation that disables guidance for t > tau*T, with tau in {0.1, 0.2, 0.5, 1.0} and all other hyperparameters fixed; report ValC, Delta MMD, and % valid graphs. If removing early guidance preserves or improves ValC and Delta MMD while increasing validity or diversity, the coarse-t Dirac approximation is misaligned and the central guidance claim fails for the early trajectory. Complement this with a denoiser error curve: for validation graphs, add noise at those same tau*T levels and measure E[||G0 - \\hat G0(Gt)||^2]/E[||G0||^2]; if the error is large exactly where the ablation shows no benefit, the mechanism is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central derivation rests on Eq. (5): the intractable conditional expectation in the Feynman-Kac control (4) is replaced by exp(-r(\\hat G0(\\hat Gt))/lambda), i.e., p(G0|\\hat Gt) is taken to be a Dirac mass at the one-step MMSE denoiser. The subsequent gradient control (6) and every non-differentiable estimator in Eqs. (9)-(12) evaluate r(.) on this same \\hat G0(\\hat Gt). Thus all claimed guidance modes share one approximation. At coarse t, the posterior p(G0|Gt) is broad and the one-step denoiser is far from the true G0; for graph rewards defined on discrete objects (thresholded adjacency, triangle counts, fairness metrics), r(\\hat G0) can be flat or sensitive to denoiser artifacts, so the greedy control direction may be misaligned and can push samples off the data manifold. The paper acknowledges this only in passing ('the impact of such approximation errors may often diminish as t -> 0', Section 3.2) but still applies guidance at every step with no safeguard or quantitative error analysis. The claimed balance of reward alignment and fidelity therefore rests on an untested assumption about the accuracy of one-step denoising across the noise schedule.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes GGDiff, a framework for conditioning pre-trained continuous graph diffusion models on arbitrary reward functions. Conditional generation is formulated as a stochastic optimal control (SOC) problem, and the intractable optimal control of Eq. (4) is approximated by a greedy scheme built on the one-step denoiser: differentiable rewards use the gradient of the reward at the denoised sample (Eq. (6)), while non-differentiable rewards use one of three zero-order estimators (Eqs. (9)-(12)). The method is evaluated on structural constraints (maximum degree, edge count, triangle count, star graphs), fair graph generation, and incomplete graph generation, with comparisons to PRODIGY on the first set of tasks.","tokens_in":18083,"tokens_out":8743,"duration_ms":91356,"significance":"If the approximation in Eq. (5) is reliable and the loss-selection protocol is controlled, GGDiff would be a practically useful contribution: it brings gradient-free guidance to graph diffusion and unifies several known guidance heuristics under one control-theoretic umbrella. The paper provides a standard SOC derivation (Appendix A), releases code, and covers a broader set of constraint types than PRODIGY, including fairness and link prediction. At the same time, the central Dirac-delta approximation is not validated at coarse noise levels, the zero-order estimators are not formally connected to the SOC solution, and several empirical claims are stronger than the reported numbers support. These issues are fixable but require substantive additional analysis.","major_comments":[{"comment":"The Dirac-delta approximation p(G0^C | \\hat G_t^C) = δ(G0^C - \\hat G0^C(\\hat G_t^C)) is the unique bridge between the exact SOC control in Eq. (4) and every implemented guidance rule, including the zero-order estimators of Eqs. (9)-(12), which all evaluate the reward at the one-step denoiser. At coarse noise levels the posterior p(G0 | G_t) is broad and the one-step denoiser is known to be inaccurate, so the control direction can be misaligned and push samples off the data manifold. The paper's only safeguard is the remark in Section 3.2 that approximation errors diminish as t approaches 0; no error bound, no ablation of the guidance starting time, and no comparison with a multi-step denoising estimate are provided. Because this assumption is inherited by all GGDiff variants, the claimed general balance of reward alignment and fidelity is not established.","section":"Section 3.2, Eq. (5)"},{"comment":"The zero-order guidance is introduced as the argmax in Eq. (7) over directions U_t of r(\\hat G0^C(\\hat G_t^C + μ U_t)), but no derivation connects this to the optimal control in Eq. (4) or to the gradient approximation in Eq. (6). In particular, the best-of-N rule in Eq. (11) selects a direction that maximizes the reward after perturbation without any gradient magnitude, expectation over the uncontrolled process, or path-integral interpretation. The text acknowledges these are 'pseudo-gradients,' but the paper's central claim of unifying gradient-based, control-based, and zero-order guidance under a single SOC formulation requires at least a formal statement of the sense in which Eq. (7) approximates Eq. (4). Without this, the theoretical contribution is a collection of heuristics rather than a unified framework.","section":"Section 3.2.2, Eqs. (7)-(12)"},{"comment":"The manuscript states that the loss function for each constraint is 'empirically selected from a pool of possibilities based on which yields the best performance.' Since this selection is made after seeing the evaluation metrics (ValC and ΔMMD) on the test sets, the results in Table 1 are at risk of selection on the test set, and they do not support the zero-shot, plug-and-play claim for arbitrary rewards. The authors should either specify a fixed selection protocol using a validation split and report the number and identity of candidate losses, or reframe the method as requiring per-task tuning. As written, the reported ValC/ΔMMD numbers conflate the algorithm with its tuning.","section":"Appendix E.3.1 (and Table 1)"},{"comment":"The abstract and Section 4 claim that GGDiff achieves 'superior alignment with target rewards while maintaining diversity and fidelity,' but Table 1 reports negative ΔMMD values for most GGDiff cells, e.g., GGDiff-C on Community Small Max Degree (−0.73 with ValC 1.00), GGDiff-G on Enzymes Max Degree (−0.37), and GGDiff-Z on Ego Small Triangle Count (−0.07). Since the unconstrained baseline is defined as 0.00, these values indicate that the generated graphs deviate more from the original data distribution than the unconstrained model, and in several cells PRODIGY has a better ΔMMD (e.g., PRODIGY 0.07 vs. GGDiff-G −0.37 on Enzymes Max Degree). The paper needs to acknowledge this trade-off explicitly and report combined statistics or a Pareto-style evaluation; the fidelity claim is contradicted by the paper's own headline table.","section":"Table 1 and Section 4.1"},{"comment":"The fairness and incomplete-graph experiments compare only against the unconstrained model and not against any prior conditional graph generation baseline. Since the introduction states that GGDiff 'beats current state-of-the-art architectures' and achieves 'superior performance and flexibility over existing methods,' the absence of competitive baselines in two of the three experimental areas leaves the state-of-the-art claim unsupported outside the PRODIGY comparison in Section 4.1. The authors should add at least one baseline per task (e.g., classifier guidance on DiGress or an adapted PRODIGY variant where possible) or restrict the claims to what the experiments actually compare.","section":"Sections 4.2 and 4.3"}],"minor_comments":[{"comment":"The boundary condition below Eq. (14) states V0(G0) = r(G0), but Eq. (13) defines the value function as an expectation involving −r(G0), which implies V0(G0) = −r(G0). This sign inconsistency should be corrected in the derivation.","section":"Appendix A, Eq. (16)"},{"comment":"The text says the variance of the one-point estimator 'explodes as µ increases to 0'; this should read 'as µ decreases to 0' or 'as µ → 0'.","section":"Section 3.2.2, after Eq. (9)"},{"comment":"The notation U_t^{(i)} = argmax_{...} r(...) · U_t is not well-formed because the argmax returns an index, not a direction. Define i* = argmax_i r(\\hat G0^C(\\hat G_t^C + μ U_t^{(i)})) and then set U_t = k U_t^{(i*)}.","section":"Eq. (11)"},{"comment":"Algorithm 1 computes U_t = ∇_{\\hat G_t} r(\\hat G0^C(\\hat G_t^C)), but the derivation in Eq. (6) includes the factor g(t)/λ. The implementation should state how the step size k, g(t), and λ are combined, or the algorithm should match the equation.","section":"Algorithm 1, lines 5-6"},{"comment":"The values of the key hyperparameters N, k, μ, λ, and the guidance schedule are not reported anywhere in the main text or appendices; since the method's behavior is sensitive to these quantities, they should be listed per dataset and task to make the experiments reproducible.","section":"Appendix E"}],"recommendation":"major_revision","confidential_remarks":"The fair-graph experiments use the fairness loss and metrics from Navarro et al. [21], a paper co-authored by two of the present authors, and the evaluation uses the same type of metric that the loss is designed to improve. This is a mild self-reference and, by itself, not disqualifying, but it reinforces the need for baselines from independent groups. The empirical loss selection in Appendix E.3.1 is a more serious concern because it resembles test-set tuning; the editor may wish to ask for a clarification of the selection protocol in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful application of existing SOC guidance ideas to graph diffusion, and the best-of-N zero-order variant is a sensible practical contribution. The paper is not as novel or as tightly derived as its abstract suggests, but it is honest about its main heuristic and the experiments, while incomplete, show real promise. I'd send it to review, with the expectation of major revision.\n\nWhat is actually new: it is the first to port SOC-based guidance to continuous graph diffusion, and it makes a fair point that PRODIGY-style projection needs closed-form operators while ZO evaluation does not. The framework naturally recovers gradient guidance as the differentiable case and offers three ZO estimators; the greedy best-of-N direction is a good fit for discrete graph rewards. The star-graph experiment nicely exposes that PRODIGY collapses to single-node graphs, which is a useful negative result. Credit where due: the paper states in Section 3.2.2 that the ZO estimator is a surrogate direction, not an unbiased gradient estimate, and Appendix E admits the loss functions were empirically selected per task. That is more transparent than most papers in this area.\n\nWhere it is soft. The whole guidance family inherits the Dirac-delta approximation at the one-step denoiser, Eq. (5). The paper acknowledges this only in passing and provides no ablation or error analysis at coarse noise levels. That said, this is the same approximation DPS and others use, so I don't read it as fatal; it does mean the 'unification' is really a family of heuristics sharing one approximation. More important, Eq. (7) is asserted rather than derived from Eq. (4); connecting the ZO greedy search to the Feynman-Kac control would make the framework coherent. On the experimental side: no code, no hyperparameters for mu, k, N, lambda, and the per-task loss selection post hoc means the reported numbers are tuned. On fair graph generation and link prediction there is no non-trivial baseline, only unconstrained generation, so 'superior' is not established there. The self-citation in the fairness section is to a metric the authors co-developed; it is mild and not a real problem.\n\nWho is this for: people working on graph diffusion conditioning and guided generation in drug design or fair graph generation. It deserves a serious referee, but the revision needs to add a derivation or at least a careful justification of the ZO step, a code/data release, fixed hyperparameters, and a stronger baseline comparison. I would not desk-reject it.","headline":"A useful, honestly-reported application of SOC guidance to graph diffusion; the best-of-N zero-order variant is a practical contribution, but the unification claim and superiority claims need a sharper derivation, code, and stronger baselines before they fully hold.","tokens_in":18578,"tokens_out":2427,"would_cite":false,"duration_ms":24226,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes GGDiff, which treats conditional graph diffusion as a stochastic optimal control problem and approximates the optimal control with one-step denoising plus zero-order search, enabling guidance of pre-trained models by…","keywords":["graph generation","diffusion models","conditional generation","stochastic optimal control","zero-order optimization","non-differentiable rewards","graph guidance","fairness constraints"],"falsifier":"Run GGDiff on the ego-small dataset with a deliberately corrupted one-step denoiser active only in the first half of the reverse process: if reward alignment and validity stay unchanged while the corruption is active, the Dirac-delta approximation is not load-bearing; if they degrade sharply, the central claim depends on the denoiser being accurate at coarse noise levels.","tokens_in":17580,"feed_emoji":"🕸️","tokens_out":7273,"duration_ms":58286,"temperature":0.7,"pith_summary":"The paper proposes GGDiff, a framework for conditional graph generation that works with arbitrary reward functions, including non-differentiable ones. Its central claim is that guiding a pre-trained continuous graph diffusion model can be treated as a stochastic optimal control problem, and that the resulting optimal control can be made tractable by evaluating the reward on a one-step denoised estimate of the graph plus randomized-direction search. If correct, GGDiff would let users impose structural constraints, fairness criteria, or partial-observation conditions on generated graphs without retraining the model, computing gradients of the reward, or designing projection operators. Experiments on motif constraints, fair graph generation, and link prediction report better reward alignment than a projection-based baseline while preserving the distribution of the underlying graph family.","feed_headline":"One framework steers graph diffusion with black-box rewards","feed_subtitle":"A stochastic-control trick makes non-differentiable rewards usable for steering graph diffusion.","key_machinery":"The load-bearing object is the optimal control law $U^*(G_t^C,t) = -g(t)\\nabla_{G_t^C}\\log \\mathbb{E}_{\\mathrm{pre}}[\\exp(-r(G_0^C)/\\lambda)\\mid G_t^C]$ from the Feynman-Kac solution of the linearized Hamilton-Jacobi-Bellman equation. Since the expectation is intractable, the paper applies a greedy approximation: $p(G_0^C\\mid G_t^C)$ is treated as a Dirac delta at the one-step MMSE denoiser $\\hat{G}_0^C(G_t^C)$ from Tweedie's formula, turning the control into the reward gradient $\\frac{g(t)}{\\lambda}\\nabla_{\\hat{G}_t^C}r(\\hat{G}_0^C(\\hat{G}_t^C))$. For non-differentiable rewards, the gradient is replaced by a general zero-order estimator $\\hat{\\nabla}r(\\hat{G}_t^C):=\\mathbb{E}_{U_t\\sim D}[w(U_t)r(\\hat{G}_0^C(\\hat{G}_t^C+\\mu U_t))U_t]$, instantiated as one-point, two-point, best-of-N, and multi-point versions. The one-step denoiser is what connects the reward (defined on clean graphs) to the noisy state, and all guidance rules inherit their quality from it.","core_discovery":"On the paper's own terms, the discovery is that conditional graph generation should be posed as a stochastic control problem: the sampler follows a controlled diffusion whose drift contains an extra control term, and the optimal control is given by the Feynman-Kac formula as the gradient of a log-expectation of the exponentiated reward. Because that expectation is intractable, the paper replaces the conditional distribution of the clean graph with a Dirac delta at the one-step Tweedie denoiser, yielding a greedy control proportional to the gradient of the reward at the denoised estimate; for non-differentiable rewards it replaces the gradient with zero-order estimators—one-point, two-point, best-of-N, and multi-point random search—that evaluate the reward along random directions. The paper argues this single formulation unifies gradient-based guidance, control-based guidance, and gradient-free optimization, and that it enables zero-shot guidance of pre-trained diffusion models under both differentiable and black-box rewards.","pith_inferences":["Because the greedy Dirac-delta approximation is known by the authors to degrade away from $t=0$, a natural extension is an adaptive schedule that throttles guidance strength or uses multi-step denoisers early in the reverse process; the paper does not test this.","The same control-plus-zero-order recipe is not graph-specific once a denoiser and a reward are available, so GGDiff's machinery could plausibly transfer to other structured outputs such as point clouds, sets, or trajectories.","A direct comparison between GGDiff's greedy control and an exact Monte Carlo evaluation of the Feynman-Kac expectation on small graphs would isolate how much performance comes from the control formulation versus from the one-step approximation; the paper does not run this comparison.","The best-of-N estimator's bias toward high-reward directions suggests GGDiff may trade away some modes of the prior when rewards are multimodal; testing diversity under multi-peaked rewards would clarify that trade-off."],"forward_implications":["A user can condition a pre-trained graph diffusion model on a black-box reward with no retraining, no reward gradient, and no closed-form projection operator.","Differentiable rewards automatically recover the standard gradient-guidance rule used in diffusion posterior sampling, so the framework contains existing gradient guidance as a special case.","Non-differentiable constraints become expressible directly—for example, quantized adjacency matrices or one-sided penalties—expanding the space of graph conditions that can be targeted.","Empirically, the framework reports higher constraint-satisfaction rates and better distribution preservation than the projection-based baseline on degree, edge-count, and triangle constraints, and it handles fairness and partial-observation tasks that the baseline cannot.","Because all guidance is computed through reward evaluations on the one-step denoiser, the method's computational cost is controlled by the number of reward queries per diffusion step, with best-of-N giving low variance in discrete settings."],"supporting_citations":[{"why":"Defines the continuous graph diffusion SDE and score networks (GDSS) that serve as the pretrained prior GGDiff steers.","marker":"[12]"},{"why":"Provides the DDPM sampler and update equations used in Algorithm 1 for the reverse process.","marker":"[8]"},{"why":"Establishes the diffusion posterior sampling gradient guidance that GGDiff's differentiable control reduces to for log-likelihood rewards.","marker":"[5]"},{"why":"Is the projection-based conditional graph generation baseline whose constraints and experimental setup GGDiff matches and outperforms.","marker":"[28]"},{"why":"Shows non-differentiable rule-guided diffusion in another domain, which GGDiff's control formulation generalizes.","marker":"[11]"},{"why":"Supplies the zero-order optimization background and estimator framework used for non-differentiable rewards.","marker":"[15]"},{"why":"Provides the mean-squared-error analyses of the zero-order estimators that justify the two-point and multi-point choices.","marker":"[16]"},{"why":"Defines the dyadic parity fairness metrics used to evaluate fair graph generation.","marker":"[21]"},{"why":"Provides the stochastic control and Feynman-Kac results used to derive the optimal control law.","marker":"[25]"}],"fun_headline_variants":["Graph diffusion guided by stochastic control, not just gradients","Unified guidance for graph generation: gradients and black-box rewards","Zero-shot control of graph diffusion with any reward","Stochastic control makes graph diffusion steerable by any reward","GGDiff: one recipe to guide graph diffusion with any reward"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The guidance direction is computed by treating the one-step denoised graph as if it were the true final graph (a Dirac-delta approximation), so if that denoiser is inaccurate—especially early in the reverse process—the reward signal is misaligned and the steering can push samples off the data manifold.","fun_headline_variants_meta":{"raw":{"variants":["Graph diffusion guided by stochastic control, not just gradients","Unified guidance for graph generation: gradients and black-box rewards","Zero-shot control of graph diffusion with any reward","Stochastic control makes graph diffusion steerable by any reward","GGDiff: one recipe to guide graph diffusion with any reward"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000757,"raw_usage":{"total_tokens":3364,"prompt_tokens":941,"completion_tokens":2423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2343}},"tokens_in":557,"tokens_out":2423,"duration_ms":14504,"temperature":1.0,"reasoning_tokens":2343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:08:28.526931+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run GGDiff on the ego-small dataset with a deliberately corrupted one-step denoiser active only in the first half of the reverse process: if reward alignment and validity stay unchanged while the corruption is active, the Dirac-delta approximation is not load-bearing; if they degrade sharply, the central claim depends on the denoiser being accurate at coarse noise levels.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the continuous graph diffusion SDE and score networks (GDSS) that serve as the pretrained prior GGDiff steers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the DDPM sampler and update equations used in Algorithm 1 for the reverse process."},{"cited_title":"T., Klasky, M","cited_arxiv_id":null,"evidence_quote":"Establishes the diffusion posterior sampling gradient guidance that GGDiff's differentiable control reduces to for log-likelihood rewards."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the projection-based conditional graph generation baseline whose constraints and experimental setup GGDiff matches and outperforms."},{"cited_title":"O., and Varshney, P","cited_arxiv_id":null,"evidence_quote":"Supplies the zero-order optimization background and estimator framework used for non-differentiable rewards."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the mean-squared-error analyses of the zero-order estimators that justify the two-point and multi-point choices."},{"cited_title":"G., and Segarra, S","cited_arxiv_id":null,"evidence_quote":"Defines the dyadic parity fairness metrics used to evaluate fair graph generation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the stochastic control and Feynman-Kac results used to derive the optimal control law."}],"review_version":1}