H-GLow: Accelerating Graph Generation via Hazard-Guided Discrete Flow Matching

Sirine Ayadi; Stephan Günnemann

cs.LGOct 24, 2026

H-GLow: Accelerating Graph Generation via Hazard-Guided Discrete Flow Matching

Authors: Sirine Ayadi, Stephan Günnemann

Organizations: School of Computation, Information and Technology, Technical University of Munich · Munich Data Science Institute & Munich Center for Machine Learning

Abstract

Discrete flow matching (DFM) offers a promising approach to graph generation; however, generating high-quality graphs with few steps remains challenging due to the strong dependencies between nodes and edges. Although DFM decouples training from sampling, enabling flexible sampling strategies without retraining, existing graph DFM samplers often rely on task-specific heuristics, substantially expanding the hyperparameter space. Moreover, graph DFM models still require sampling budgets comparable to those of diffusion-based approaches, leaving their potential for accelerating graph generation only partially realized. In this work, we introduce H-Glow, a hazard-guided framework for accelerating graph generation built on the decomposition of Continuous-Time Markov chain (CTMC) generators into hazards and conditional jump distributions. H-Glow leverages these quantities to plan graph edits and dynamically adjust the sampling time to graph activity. H-Glow can be adapted to pretrained graph DFM models, substantially improving the generation quality with fewer steps. We further introduce a novel graph DFM model based on the conditional generator matching framework, which directly predicts hazards and conditional jumps, and show that its training objective decomposes exactly into hazard matching and hazard-weighted jump prediction. Experiments on molecular and synthetic graph benchmarks demonstrate improved generation quality with substantially fewer sampling steps.

Figures & tables

Figure 1: Overview of H-Glow.

Model	Steps (NFE)	Valid $\uparrow$	Relaxed Validity $\uparrow$	Unique $\uparrow$	FCD $\downarrow$
Training set	–	$99.3$	$99.5$	$99.2$	$0.03$
DiGress ( Vignac et al., 2022 )	500 (500 $\pm$ 0.0)	$99.0$	–	$96.2$	–
CatFlow ( Eijkelboom et al., 2024 )	–	$99.8$	–	$100.0$	$0.44$
DisCo ( Xu et al., 2024 )	500 (500 $\pm$ 0.0)	$99.3$	–	–	–
Cometh ( Siraudin et al., 2024 )	500 (500 $\pm$ 0.0)	$99.6$	–	$96.8$	$0.25$
SID ( Boget, 2025 )	500 (500 $\pm$ 0.0)	$99.7$	–	–	$0.50$

Table 1: Performance on QM9. We report Steps , denoting the available sampling budget

T

, and the mean

\pm

standard deviation of NFE across sampled molecules.

	MOSES								GuacaMol
Model	Steps (NFE)	Val. $\uparrow$	Unique. $\uparrow$	Novelty $\uparrow$	Filters $\uparrow$	FCD $\downarrow$	SNN $\uparrow$	Scaf. $\uparrow$	Steps (NFE)	Val. $\uparrow$	V.U. $\uparrow$	V.U.N. $\uparrow$	KLD. $\uparrow$	FCD $\uparrow$
Training set	–	$100.0$	$100.0$	$0.0$	$100.0$	$0.01$	$0.64$	$99.1$	–	$100.0$	$100.0$	$0.0$	$99.9$	$92.8$
DiGress ( Vignac et al., 2022 )	500 (500 $\pm$ 0.0)	$85.7$	$100.0$	$95.0$	$97.1$	$1.19$	$0.52$	$14.8$	500 (500 $\pm$ 0.0)	$85.2$	$85.2$	$85.1$	$92.9$	$68.0$
DisCo ( Xu et al., 2024 )	500 (500 $\pm$ 0.0)	$88.3$	$100.0$	$97.7$	$95.6$	$1.44$	$0.50$	$15.1$	500 (500 $\pm$ 0.0)	$86.6$	$86.6$	$86.5$	$92.6$	$59.7$
Cometh ( Siraudin et al., 2024 )	500 (500 $\pm$ 0.0)	$86.1$	$100.0$	$96.0$	$97.3$	$1.27$	$0.50$	$16.4$	500 (500 $\pm$ 0.0)	$94.4$	$94.4$	$93.5$	$94.1$	$67.4$
Cometh-PC ( Siraudin et al., 2024 )	500 (500 $\pm$ 0.0)	$90.5$	$99.9$	$92.6$	$99.1$	$1.27$	$0.54$	$16.0$	500 (500 $\pm$ 0.0)	$98.9$	$98.9$	$97.6$	$96.7$	$72.7$

Table 2: Large molecule generation performance on MOSES and GuacaMol. Only iterative denoising-based methods are reported. We report Steps , denoting the sampling budget

T

, and the mean

\pm

standard deviation of NFE across sampled molecules.

		Planar		Tree
Model	Steps	V.U.N. $\uparrow$	Ratio $\downarrow$	V.U.N. $\uparrow$	Ratio $\downarrow$
Training set	–	$100.0$	$1.0$	$100.0$	$1.0$
EDGE ( Chen et al., 2023 )	1000	$0.0$	$431.4$	$0.0$	$850.7$
DiGress ( Vignac et al., 2022 )	1000	$77.5$	$5.1$	$90.0$	$1.6$
HSpectre ( Bergmeister et al., 2024 )	–	$95.0$	$2.1$	$100.0$	$4.0$
GruM ( Jo et al., 2023 )	–	$90.0$	$1.8$	–	–

Table 3: Performance on synthetic graphs. Full version in Table 13 .

Figure 2: Sampler comparison using the same pretrained DeFoG model on GuacaMol and MOSES. We compare base parallel Euler sampling (blue), Euler sampling with DeFoG’s added target guidance and stochasticity

(\eta,\omega)

(purple), and our H-Glow sampler applied to DeFoG’s base generator (green). Our H-Glow sampling strategy achieves high sample quality with substantially fewer sampling steps.

	$T=50$		$T=100$
Clock	V.U.N. $\uparrow$	Ratio $\downarrow$	V.U.N. $\uparrow$	Ratio $\downarrow$
Hazard-guided	99.0	1.86	99.0	1.81
Uniform	99.5	23.43	100.0	49.11
Polydec-2	100.0	20.20	100.0	42.62
Polydec-10	99.5	5.57	99.5	12.66
Polydec-20	95.5	3.88	99.5	6.08

Table 4: Clock ablations.

	$T=50$		$T=100$
Variant	Val. $\uparrow$	FCD $\downarrow$	Val. $\uparrow$	FCD $\downarrow$
H-Glow	89.5	0.98	89.9	0.96
Hazard-only	88.1	1.10	89.2	1.12
w/o early stopping	88.1	1.41	88.7	1.82
w/o hazard annealing	88.5	1.43	87.9	1.39

Table 5: Design choices.

	$T=50$		$T=100$
Sampler	Val. $\uparrow$	FCD $\downarrow$	Val. $\uparrow$	FCD $\downarrow$
H-Glow	89.5	0.98	89.9	0.96
Euler	80.3	1.38	85.8	1.38
RVF	86.4	1.03	87.7	1.08

Table 6: Sampler comparison.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	# Graphs	Train/Val/Test	$E$	$X$	Avg. $B$	Max. $B$	Avg. $N$	Max. $N$
QM9	130,831	97,734/20,042/13,055	4	4	18.9	28	8.8	9
MOSES	1,936,962	1,584,663/176,225/176,074	4	8	46.5	62	21.7	27
GuacaMol	1,398,213	1,118,633/69,926/209,654	4	12	60.4	176	27.8	88
Planar	200	128/32/40	1	1	355.7	362	64.0	64
Tree	200	128/32/40	1	1	126.0	126	64.0	64

Appendix

Table 7: Dataset statistics.

E

and

X

denote the numbers of present-edge and node categories, respectively. The no-edge category is excluded from

E

B

counts directed edge entries, following the convention of prior work, while

N

denotes the number of nodes.

	QM9	GuacaMol	MOSES	Planar	Tree
Epochs	1000	250	300	100000	100000
Learning rate	$2\times 10^{-4}$	$2\times 10^{-4}$	$2\times 10^{-4}$	$2\times 10^{-4}$	$2\times 10^{-4}$
Optimizer	AdamW	AdamW	AdamW	AdamW	AdamW
AMSGrad	Yes	Yes	Yes	Yes	Yes
RRWP steps	12	20	20	32	128
Number of layers	9	12	12	10	10

Appendix

Table 8: Overview of the training hyperparameters of H-Glow.

		$T=20$		$T=50$		$T=100$
Model	Sampler	Val. $\uparrow$	FCD $\downarrow$	Val. $\uparrow$	FCD $\downarrow$	Val. $\uparrow$	FCD $\downarrow$
Denoiser	Euler	97.0	0.75	98.9	0.28	99.4	0.14
H-Glow	Euler	96.7	0.66	99.2	0.24	99.5	0.13
Denoiser	H-Glow	98.9	0.31	99.0	0.21	99.0	0.11
H-Glow	H-Glow	99.2	0.37	99.1	0.10	99.2	0.10

Appendix

Table 9: Parameterization comparison.

Dataset	Steps $T$	NFE
QM9	$20$	19.9 ${}\pm{}$	0.4
QM9	$50$	22.4 ${}\pm{}$	2.3
MOSES	$50$	35.9 ${}\pm{}$	7.0
MOSES	$100$	47.1 ${}\pm{}$	15.7
MOSES	$200$	57.2 ${}\pm{}$	14.2
GuacaMol	$50$	40.4 ${}\pm{}$	8.1

Appendix

Table 10: NFE across datasets and sampling steps.

Figure 3: Schedulers.

$T$	Sampler	Validity $\uparrow$	Unique $\uparrow$	Novelty $\uparrow$	Filters $\uparrow$	FCD $\downarrow$
50	RVF (default)	86.4	100.0	94.6	99.1	1.03
50	DeFoG (default)	80.3	100.0	95.3	98.0	1.38
50	HGLow (ours)	89.5	100.0	94.4	99.1	0.98
100	RVF (default)	87.7	100.0	94.2	99.2	1.08
100	DeFoG (default)	85.8	100.0	90.7	99.4	1.38
100	HGLow (ours)	89.9	99.9	93.0	99.2	0.96

Appendix

Table 11: Sampler comparison using the HGLow trained model on MOSES.

			MOSES							GuacaMol
Model	Class	Steps	Val. $\uparrow$	Unique. $\uparrow$	Novelty $\uparrow$	Filters $\uparrow$	FCD $\downarrow$	SNN $\uparrow$	Scaf. $\uparrow$	Val. $\uparrow$	V.U. $\uparrow$	V.U.N. $\uparrow$	KLD. $\uparrow$	FCD $\uparrow$
Training set	–	–	$100.0$	$100.0$	$0.0$	$100.0$	$0.01$	$0.64$	$99.1$	$100.0$	$100.0$	$0.0$	$99.9$	$92.8$
DiGress ( Vignac et al., 2022 )	Diff.	500	$85.7$	$100.0$	$95.0$	$97.1$	$1.19$	$0.52$	$14.8$	$85.2$	$85.2$	$85.1$	$92.9$	$68.0$
DisCo ( Xu et al., 2024 )	Diff.	500	$88.3$	$100.0$	$97.7$	$95.6$	$1.44$	$0.50$	$15.1$	$86.6$	$86.6$	$86.5$	$92.6$	$59.7$
Cometh ( Siraudin et al., 2024 )	Diff.	500	$86.1$	$100.0$	$96.0$	$97.3$	$1.27$	$0.50$	$16.4$	$94.4$	$94.4$	$93.5$	$94.1$	$67.4$
Cometh-PC ( Siraudin et al., 2024 )	Diff.	500	$90.5$	$99.9$	$92.6$	$99.1$	$1.27$	$0.54$	$16.0$	$98.9$	$98.9$	$97.6$	$96.7$	$72.7$

Appendix

Table 12: Large molecule generation performance on MOSES and GuacaMol.

			Planar		Tree
Model	Class	Steps	V.U.N. $\uparrow$	Ratio $\downarrow$	V.U.N. $\uparrow$	Ratio $\downarrow$
Training set	–	–	$100.0$	$1.0$	$100.0$	$1.0$
GraphRNN ( You et al., 2018 )	AR	–	$0.0$	$490.2$	$0.0$	$607.0$
GRAN ( Liao et al., 2019 )	AR	–	$0.0$	$2.0$	$0.0$	$607.0$
BiGG ( Dai et al., 2020 )	AR	–	$5.0$	$16.0$	$75.0$	$5.2$
GraphGen ( Goyal et al., 2020 )	AR	–	$7.5$	$210.3$	$95.0$	$33.2$

Appendix

Table 13: Performance on the synthetic datasets Planar and Tree. AR denotes Autoregressive approaches.

Panel of Figure 4 — Figure 4: Uncurated sets of molecular graphs generated by H-Glow for the MOSES dataset for $T\in\{50,100,200\}$ steps.

Panel of Figure 5 — Figure 5: Uncurated sets of molecular graphs generated by H-Glow for the Guacamol dataset for $T\in\{50,100,200\}$ steps.

Panel of Figure 6 — Figure 6: Uncurated sets of molecular graphs generated by H-Glow for the QM9 dataset for $T\in\{20,50\}$ steps.

Panel of Figure 7 — Figure 7: Uncurated sets of planar graphs generated by H-Glow for the Planar dataset for $T\in\{20,50,100,200\}$ steps.

Graph Generation

Graph generation builds models and algorithms that create graphs with desired structures, properties, or distributions.

See all 67 papers

Hierarchical Discrete Flow Matching for Graph Generation
Mar 31, 2026Yoann Boget, Pablo Strasser, Alexandros KalousisGraph Generation Flow-Based Generative Modeling
Department of Computer Science, University of Geneva, Switzerland · DMML Group, Geneva School for Business administration HES-SO, Switzerland
Efficient Graph Generation via Direct Prediction and Flow Matching
Oct 4, 2026Susie LuFlow Matching Graph Generation
MIT
Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation
Mar 24, 2026Michal Balcerak, Suprosanna Shit, Chinmay Prabhakar +4Generative Modeling Energy-Based Models
University of Zurich · Harvard University · Kempner Institute

Mar 31, 2026J/K move · Enter open · S save

Hierarchical Discrete Flow Matching for Graph Generation

Yoann Boget, Pablo Strasser, Alexandros Kalousis

Department of Computer Science, University of Geneva, Switzerland · DMML Group, Geneva School for Business administration HES-SO, Switzerland

Graph Generation Flow-Based Generative Modeling Discrete Flow Matching

Open paper AskPDF arXiv

Denoising-based models, including diffusion and flow matching, have led to substantial advances in graph generation. Despite this progress, such models remain constrained by two fundamental limitations: a computational cost that scales quadratically with the number of nodes and a large number of function evaluations required during generation. In this work, we introduce a novel hierarchical generative framework that reduces the number of node pairs that must be evaluated and adopts discrete flow matching to significantly decrease the number of denoising iterations. We empirically demonstrate that our approach more effectively captures graph distributions while substantially reducing generation time.

Abstract

Graph Generation

Explore similar work

Hierarchical Discrete Flow Matching for Graph Generation

Efficient Graph Generation via Direct Prediction and Flow Matching

Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation