Efficient Graph Generation via Direct Prediction and Flow Matching

Susie Lu

cs.LGOct 4, 2026

Efficient Graph Generation via Direct Prediction and Flow Matching

Authors: Susie Lu

Organizations: MIT

Abstract

Generative modeling of graph-structured data is crucial for tasks ranging from drug discovery to social network simulation. Among these models, denoising diffusion models have achieved great success in graph generation by learning to progressively reverse a process that adds noise to the original graph. However, the standard noise-prediction approach of diffusion models is suboptimal for graph data. The goal for a graph generative model is to learn the clean graphs' topological properties, such as connectivity and degree distribution. Because a diffusion model that predicts noise does not explicitly learn these topological properties, it is challenging for the model to output graphs with the desired structural statistics. To address this challenge, we introduce Direct Graph Flow Matching (DiGFM), a novel graph transformer model guided by two goals: predict clean graphs and improve sampling efficiency. Distinct from the prevailing diffusion approach, DiGFM employs a continuous flow-matching paradigm and integrates direct graph prediction. Specifically, DiGFM maps the prior noise distribution to the clean graph distribution via a multi-step process: the model repeatedly predicts the underlying clean graph, and a transformation is employed to convert the model output to the velocity vector that points in the direction toward the clean graph distribution. This design enables DiGFM to generate high-quality samples using only 2.5% to 15.6% of the steps required by diffusion-based models, which leads to a 5.3x to 257x speedup in wall-clock inference time. Experiments demonstrate that DiGFM outperforms or matches prior state-of-the-art models across general graph benchmarks and molecular datasets, generating graphs with strong adherence to ground-truth structural statistics at significantly faster inference speeds.

Figures & tables

Framework	Graph Models	Pred. Space	Predicts Clean Graph?
Graph Space Models
DDPM	ConGress [ 25 ]	Noise	✗
Score-based	EDP-GNN [ 26 ] , GDSS [ 27 ]	Score	✗
Diffusion Bridge	Bridge [ 28 ]	Drift	✗
Diffusion Mixture	GruM [ 10 ]	Graph	✓
Flow-based	DiGFM (ours)	Graph	✓

Table 1: Prediction spaces of continuous graph diffusion/flow models.

Framework	Graph Models	Pred. Space	Predicts Clean Graph?
Graph Space Models
Discrete DDPM (discrete $t$ )	DiGress [ 25 ]	Graph	✓
Discrete DDPM (continuous $t$ )	DisCo [ 33 ]	Graph	✓
Discrete Flow-based	DeFoG [ 34 ]	Graph	✓

Table 2: Prediction spaces of discrete graph diffusion/flow models.

Figure 1: Visualization of a single graph generated by DiGFM on each dataset.

	Community (Synthetic)					Ego (Real)
Model	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	FID $\downarrow$	RBF $\downarrow$	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	FID $\downarrow$	RBF $\downarrow$
Training set	0.0028	0.0128	0.0129	2.0239	0.0134	0.0002	0.0078	0.0060	0.4216	0.0093
GraphRNN [ 13 ]	0.1440	0.0535	0.0198	8.3869	0.1591	0.0768	1.1456	0.1087	90.5655	0.6827
GRAN [ 40 ]	0.1022	0.0894	0.0198	64.115	0.0749	0.5778	0.3360	0.0406	489.960	0.2633
GraphCNF [ 41 ]	0.1129	1.2882	0.0197	29.153	0.1341	0.1010	0.7654	0.0820	18.7929	0.0896
GDSS [ 27 ]	0.0535	0.2072	0.0196	6.5531	0.0443	0.8189	0.6032	0.3315	60.6100	0.4331

Table 3: Generation results on Community and Ego graphs. DiGFM outperforms baselines on the majority of metrics, while only using up to 10% of their sampling steps.

	SBM (Synthetic)					Enzymes (Real)
Model	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Spec. $\downarrow$	V.U.N. $\uparrow$	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Avg. $\downarrow$
Training set	0.0008	0.0332	0.0255	0.0063	0	0.003	0.020	0.017	0.014
GraphRNN [ 13 ]	0.0055	0.0584	0.0785	0.0065	5	0.017	0.062	0.046	0.042
SPECTRE [ 15 ]	0.0015	0.0521	0.0412	0.0056	53	-	-	-	-
EDP-GNN [ 26 ]	0.0011	0.0552	0.0520	0.0070	35	0.023	0.268	0.082	0.124
GDSS [ 27 ]	0.0212	0.0646	0.0894	0.0128	5	0.026	0.061	0.009	0.032

Table 4: Generation results on SBM and Enzymes graphs. DiGFM outperforms baselines on the majority of metrics, while only using up to 10% of their sampling steps.

	QM9 (Real)				ZINC250k (Real)
Model	Valid % $\uparrow$	FCD $\downarrow$	NSPDK $\downarrow$	Scaf. $\uparrow$	Valid % $\uparrow$	FCD $\downarrow$	NSPDK $\downarrow$	Scaf. $\uparrow$
Training set	100	0.040	0.0001	0.9719	100	0.062	0.0001	0.8395
MoFlow [ 44 ]	91.36	4.467	0.0169	0.1447	63.11	20.931	0.0455	0.0133
GraphAF [ 45 ]	74.43	5.625	0.0207	0.3046	68.47	16.023	0.0442	0.0672
GraphDF [ 46 ]	93.88	10.93	0.0636	0.0978	90.61	33.546	0.1770	0.0000
EDP-GNN [ 26 ]	47.52	2.680	0.0046	0.3270	82.97	16.737	0.0485	0.0000

Table 5: Generation results on molecule datasets. DiGFM achieves competitive performance on both QM9 and ZINC250k datasets, while only using up to 15.6% of their sampling steps.

		Dipole Moment ( $\mu$ )		HOMO		Both ( $\mu$ & HOMO)
Model	With guidance?	MAE ( $\downarrow$ )	Valid % ( $\uparrow$ )	MAE ( $\downarrow$ )	Valid % ( $\uparrow$ )	MAE ( $\downarrow$ )	Valid % ( $\uparrow$ )
DiGress [ 25 ]	No	1.68	91.5	0.95	91.5	1.32	91.5
FreeGress [ 47 ]	No	1.03	86.7	0.45	92.0	0.83	82.3
DiGFM (Ours)	No	1.67	96.4	0.26	96.1	0.60	96.7
DiGress [ 25 ]	Yes	0.80	82.5	0.61	91.2	1.11	86.7
FreeGress [ 47 ]	Yes	0.74	83.7	0.32	90.1	0.68	83.4

Table 6: Conditional generation on QM9 molecules. DiGFM outperforms baselines on generation of QM9 molecules conditioned on the dipole moment

\mu

, the Highest Occupied Molecular Orbital (HOMO), and both.

Figure 2: Sampling efficiency of DiGFM. (a) DiGFM’s average MMD (red line) outperforms the state-of-the-art (gray line) in only 15 steps. (b) The global graph metrics of Validity and RBF MMD surpass the state-of-the-art at just 55 and 20 steps, respectively.

Inference Time (sec/graph)

Speedup

Dataset

GruM

DiGress

GDSS

DisCo

CatFlow

DeFoG

DiGFM (ours)

vs. GruM

vs. DiGress

vs. GDSS

vs. DisCo

vs. CatFlow

vs. DeFoG

SBM

12.75

19.92

12.39

0.995

2.11

0.714

0.136

{94\times}

{146\times}

{91\times}

{7.3\times}

{16\times}

{5.3\times}

QM9

12.06

9.09

11.60

1.88

1.59

0.578

0.047

{257\times}

{193\times}

{247\times}

{40\times}

{34\times}

{12\times}

ZINC250k

12.59

11.34

12.36

2.37

2.47

0.835

0.073

{172\times}

{155\times}

{169\times}

{32\times}

{34\times}

{11\times}

Table 7: Inference time comparison. Time (in seconds) required to generate a single graph on an A100-SXM4-80GB GPU. DiGFM is more efficient than all six baseline models.

Prediction Target	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Spec. $\downarrow$	FID $\downarrow$	RBF $\downarrow$
Clean graph $\mathbf{x}$	0.0017	0.0641	0.0233	0.0050	1.9784	0.0212
Velocity $\mathbf{v}$	0.0022	0.0870	0.0268	0.0064	4.1700	0.0335
Noise $\bm{\epsilon}$	0.0028	0.0882	0.0262	0.0101	6.7882	0.0407

Table 8: Predicting the clean graph

\mathbf{x}

outperforms predicting the velocity

\mathbf{v}

or noise

\bm{\epsilon}

Model	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Spec. $\downarrow$	FID $\downarrow$	RBF $\downarrow$	Timing (sec/graph) $\downarrow$
Ablation on # Laplacian eigenvectors as node features
DiGFM (1 eigenvector)	0.0072	0.0393	0.0239	0.0175	9.5679	0.0349	0.0296
DiGFM (2 eigenvectors)	0.0076	0.0369	0.0119	0.0150	9.7655	0.0333	0.0298
DiGFM (3 eigenvectors)	0.0063	0.0360	0.0125	0.0143	7.2446	0.0305	0.0299
DiGFM (4 eigenvectors)	0.0044	0.0348	0.0105	0.0120	6.6211	0.0287	0.0302
Ablation on higher-order structural features

Table 9: Ablation on DiGFM architectural designs.

Figure 3: Ablation on the guidance scale

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Config	Community, Ego, Enzymes, SBM	QM9	ZINC250k
# params	7.1M	8.2M	10.8M
Depth	8	8	12
MLP dims	$X$ : 128, $E$ : 64, $y$ : 128	$X$ : 256, $E$ : 256, $y$ : 128	$X$ : 256, $E$ : 128, $y$ : 128
Hidden dims	$d_{x}$ : 256, $d_{e}$ : 64, $d_{y}$ : 64	$d_{x}$ : 256, $d_{e}$ : 128, $d_{y}$ : 64	$d_{x}$ : 256, $d_{e}$ : 64, $d_{y}$ : 64
Num heads	8	8	8
FFN dims	$X$ : 256, $E$ : 64, $y$ : 256	$X$ : 256, $E$ : 256, $y$ : 128	$X$ : 256, $E$ : 128, $y$ : 128

Appendix

Table 10: Configurations of DiGFM on each dataset.

Dataset	Min nodes	Max nodes
Community	60	160
Ego	50	399
SBM	44	187
Enzymes	10	125
QM9	1	9
ZINC250k	6	38

Appendix

Table 11: Dataset statistics.

Figure 4: Ablation on guidance scale. Optimal performance is achieved at a scale of 7 for the

\mu

-conditioned model (left), and at scales of 3 and 9 for the HOMO-conditioned model (right).

Cap on Time Step	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Spec. $\downarrow$	FID $\downarrow$	RBF $\downarrow$
0.95	0.0047	0.0334	0.0133	0.0129	5.8646	0.0276
0.99	0.0044	0.0348	0.0105	0.0120	6.6211	0.0287
0.995	0.0053	0.0341	0.0113	0.0128	6.9350	0.0307

Appendix

Table 12: Ablation for the cap on the maximum flow matching time step used when training DiGFM. DiGFM’s performance remains consistent across all three cap values.

	Inference Time (sec/graph)			Speedup
Dataset	Full DeFoG	Reduced-step DeFoG	DiGFM (ours)	vs. Full DeFoG	vs. Reduced-step DeFoG
SBM	10.86	0.714	0.136	$80\times$	${5.3\times}$
QM9	5.71	0.578	0.047	$121\times$	${12\times}$
ZINC250k	13.34	0.835	0.073	$183\times$	${11\times}$

Appendix

Table 13: Inference time comparison. Time (in seconds) required to generate a single graph on an A100-SXM4-80GB GPU. DiGFM is more efficient than both the reduced-step and full versions of DeFoG.

Figure 5: Inference time scaling comparison. DiGFM maintains a significant speed advantage over baseline models (DeFoG, DiGress, and GruM) as the number of generated nodes

N

scales up to 1000.

Figure 6: Graphs from the Enzymes test set (left) vs. samples generated by DiGFM (right).

Figure 7: Graphs from the SBM test set (left) vs. samples generated by DiGFM (right).

Figure 8: Graphs from the Community test set (left) vs. samples generated by DiGFM (right).

Figure 9: Graphs from the Ego test set (left) vs. samples generated by DiGFM (right).

Figure 10: Graphs from the QM9 test set (left) vs. samples generated by DiGFM (right).

Figure 11: Graphs from the ZINC250k test set (left) vs. samples generated by DiGFM (right).

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