Bias mitigation in graph diffusion models

Meng Yu; Kun Zhan

cs.CVApr 2, 2026

Bias mitigation in graph diffusion models

Authors: Meng Yu, Kun Zhan

Organizations: School of Information Science & Engineering, Lanzhou University

Abstract

Most existing graph diffusion models have significant bias problems. We observe that the forward diffusion's maximum perturbation distribution in most models deviates from the standard Gaussian distribution, while reverse sampling consistently starts from a standard Gaussian distribution, which results in a reverse-starting bias. Together with the inherent exposure bias of diffusion models, this results in degraded generation quality. This paper proposes a comprehensive approach to mitigate both biases. To mitigate reverse-starting bias, we employ a newly designed Langevin sampling algorithm to align with the forward maximum perturbation distribution, establishing a new reverse-starting point. To address the exposure bias, we introduce a score correction mechanism based on a newly defined score difference. Our approach, which requires no network modifications, is validated across multiple models, datasets, and tasks, achieving state-of-the-art results.Code is at https://github.com/kunzhan/spp

Figures & tables

Panel of Figure 1 — Figure 1: (a) According to the diffusion equation, we write the perturbation distribution as $q({{\bm{x}}}_{t}|{{\bm{x}}}_{0})=\mathcal{N}\left({{\bm{x}}}_{t}|u_{t}{{\bm{x}}}_{0},\sigma_{t}^{2}\mathbf{I}\right)$ .The maximum perturbation distribution in the forward diffusing is $\mathcal{N}\left({{\bm{x}}}_{T}|u_{T}{{\bm{x}}}_{0},\sigma_{T}^{2}\mathbf{I}\right)$ , but the reverse-starting point always follows $\mathcal{N}({{\bm{x}}}_{T}|\textbf{0},\mathbf{I})$ , leading to significant inconsistencies between forward diffusing and reverse sampling. In particular, we observe that ${{\bm{x}}}_{T}$ of different baselines are in the low-noise state since their signal-to-noise ratios are always greater than one. More details are provided in Appendix A . (b) and (c) Expectation of $\|{{{\bm{s}}}}_{{\bm{\theta}},t}(\cdot)\|_{2}$ during sampling (without the corrector) and training on Community-small. Due to the reverse-starting bias, there is a significant difference between forward diffusing and reverse sampling in the early stages of sampling in (b). However, after applying our approach, not only is the reverse-starting bias mitigated, but the exposure bias during the sampling process is also significantly reduced.

Panel of Figure 2 — Figure 2: (a) and (b) The $\ell_{2}$ norm of the predictions from the two score networks at different time steps. (c) The generation results of two score networks with perturbations at different time steps.

Figure 3: Score correction based on the score difference. At the reverse-sampling time $t$ , the optimal score always points to ${{\bm{X}}}_{0}$ . ${{\bm{s}}}_{{\bm{\theta}},t}(\cdot)$ points to $\gamma_{t}{{\bm{X}}}_{0}$ with some deviation (partially containing ${{\bm{X}}}_{0}$ ), while ${{\bm{s}}}_{{\bm{\psi}},t}(\cdot)$ points to $\gamma^{\prime}_{t}{{\bm{X}}}_{0}$ with larger deviation (containing little ${{\bm{X}}}_{0}$ ). The difference between predicted and pseudo scores guides the predicted score towards the optimal score. We use $\lambda$ to control the angle and $\omega$ to adjust the magnitude of the corrected score. The final corrected score flexibly approaches the optimal score within the dashed box.

Dataset	Community-small				Enzymes				Grid
Info.	Synthetic, $12\leq\|V\|\leq 20$				Real, $10\leq\|V\|\leq 125$				Synthetic, $100\leq\|V\|\leq 400$
Method	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Avg. $\downarrow$	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Avg. $\downarrow$	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Avg. $\downarrow$
GDSS-OC	0.050	0.132	0.011	0.064	0.052	0.627	0.249	0.309	0.270	0.009	0.034	0.070
GDSS-OC-S++	0.021	0.061	0.005	0.029	0.067	0.099	0.007	0.058	0.105	0.004	0.061	0.066
GDSS-WC	0.045	0.088	0.007	0.045	0.044	0.069	0.002	0.038	0.111	0.005	0.070	0.070
GDSS-WC-S++	0.019	0.062	0.004	0.028	0.031	0.050	0.003	0.028	0.105	0.004	0.061	0.057

Table 1: Generation results on the generic graph datasets (Lower is better). The results of the Enzymes dataset of GDSS are reproduced by ourselves, the results of other baselines are all from published papers, and we give detailed settings and instructions in Appendix D .

Method	QM9			ZINC250k
Method	Sampling time $\downarrow$	NSPDK MMD $\downarrow$	FCD $\downarrow$	Sampling time $\downarrow$	NSPDK MMD $\downarrow$	FCD $\downarrow$
GDSS-OC	0.73e2	0.016	4.584	0.73e3	0.047	20.53
GDSS-OC-S++	5.10	0.001	1.661	0.70e3	0.050	16.79
GDSS-WC	1.61e2	0.004	2.550	1.41e3	0.019	14.66
GDSS-WC-S++	9.25	0.001	1.661	0.98e3	0.012	12.70
HGDM-OC	0.62e2	0.005	3.164	0.76e3	0.033	21.38

Table 2: Comparison of different methods on QM9 and ZINC250k datasets.

Method	Target protein
Method	parp1	fa7	5ht1b	braf	jak2
MOOD	7.017 (± 0.428)	0.733 (± 0.141)	18.673 (± 0.423)	5.240 (± 0.285)	9.200 (± 0.524)
MOOD-S++	8.286 (± 0.214)	0.900 (± 0.068)	20.354 (± 0.672)	5.653 (± 0.073)	9.167 (± 0.067)

Table 3: Novel hit ratio (%) results (

\uparrow

$N$	Method	QM9			Community-small
$N$	Method	Val. w/o corr. $\uparrow$	NSPDK MMD $\downarrow$	FCD $\downarrow$	Deg. $\downarrow$	Clus. $\downarrow$	Orbit $\downarrow$	Avg. $\downarrow$
1000	GDSS-OC	73.5	0.015	4.584	0.050	0.132	0.011	0.064
1000	GDSS-OCS++	94.0	0.001	1.671	0.021	0.061	0.005	0.029
500	GDSS-OC	46.2	0.045	7.960	0.136	0.456	0.151	0.248
500	GDSS-OC-S++	93.9	0.001	1.665	0.029	0.142	0.008	0.060
100	GDSS-OC	37.8	0.069	9.951	0.092	0.666	0.394	0.384

Table 5: Comparison of different methods under different total sampling time steps.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Model	GDSS
Dataset	Community-small		Enzymes		Grid		QM9		ZINC250k
Type	Node	Edge	Node	Edge	Node	Edge	Node	Edge	Node	Edge
SDE	VPSDE	VPSDE	VPSDE	VESDE	VPSDE	VPSDE	VESDE	VESDE	VPSDE	VESDE
$\beta_{\min}\|\sigma_{\min}$	0.1	0.1	0.1	0.2	0.1	0.2	0.1	0.1	0.1	0.2
$\beta_{\max}\|\sigma_{\max}$	1.0	1.0	1.0	1.0	1.0	0.8	1.0	1.0	1.0	1.0
$u_{T}$	0.7596	0.7596	0.7596	1.0	0.7596	0.7788	1.0	1.0	0.7596	1.0

Appendix

Table 7: The actual parameters of the forward perturbation of GDSS.

Model	HGDM
Dataset	Community-small		Enzymes		Grid		QM9		ZINC250k
Type	Node	Edge	Node	Edge	Node	Edge	Node	Edge	Node	Edge
SDE	VPSDE	VPSDE	VPSDE	VESDE	VPSDE	VESDE	VPSDE	VESDE	VPSDE	VESDE
$\beta_{\min}\|\sigma_{\min}$	0.1	0.1	0.1	0.2	0.1	0.2	0.1	0.1	0.1	0.2
$\beta_{\max}\|\sigma_{\max}$	1.0	1.0	1.0	1.0	7.0	0.8	2.0	1.0	1.0	1.0
$u_{T}$	0.7596	0.7596	0.7596	1.0	0.1695	1.0	0.5916	1.0	0.7596	1.0

Appendix

Table 8: The actual parameters of the forward perturbation of HGDM.

Model	GSDM						MOOD
Dataset	Community-small		Enzymes		Grid		QM9		ZINC250k
Type	Node	Eigen	Node	Eigen	Node	Eigen	Node	Edge	Node	Edge
SDE	VPSDE	VPSDE	VPSDE	VPSDE	VPSDE	VPSDE	VPSDE	VESDE	VPSDE	VESDE
$\beta_{\min}\|\sigma_{\min}$	0.1	0.1	0.1	0.1	0.1	0.2	0.1	0.1	0.1	0.2
$\beta_{\max}\|\sigma_{\max}$	1.0	1.0	1.0	1.0	1.0	0.8	1.0	1.0	1.0	1.0
$u_{T}$	0.7596	0.7596	0.7596	0.7596	0.7596	0.7788	1.0	1.0	0.7596	1.0

Appendix

Table 9: The actual parameters of the forward perturbation of GSDM and MOOD.

Panel of Figure 4 — Figure 4: Generation metric responses to perturbations at different time steps for two score networks.

Model	Hyper.	Comm.	Enzymes	Grid	QM9	ZINC250k
	$M$	400	420	350	400	400
	$\lambda_{1}$	0.2	0.0008	0.06	1.19	2.5
GDSS-OC-S++	$\omega_{1}$	0.998	1.0	1.0	1.09	1.0
	$\lambda_{2}$	0	0	0	0	0
	$\omega_{2}$	1.0	1.0	1.0	1.0	1.0
	$M$	280	310	280	240	220

Appendix

Table 10: Experimental parameters for sampling methods without a corrector (OC).

Model	Hyper.	Comm.	Enzymes	Grid	QM9	ZINC250k
	$M$	400	420	350	400	400
	$\lambda_{1}$	0.2	0.0008	0.06	1.19	2.5
GDSS-WC-S++	$\omega_{1}$	0.998	1.0	1.0	1.09	1.0
	$t_{c}$	0.2	0.45	0.055	0.1	0.4
	$\lambda_{2}$	0	0	0	0	0
	$\omega_{2}$	1.0	1.0	1.0	1.0	1.0

Appendix

Table 11: Experimental parameters for sampling methods with a corrector (WC).

Dataset	Community-small				Enzymes				Grid
Info.	Synthetic, $12\leq\|V\|\leq 20$				Real, $10\leq\|V\|\leq 125$				Synthetic, $100\leq\|V\|\leq 400$
Method	Deg.	Clus.	Orbit	Avg.	Deg.	Clus.	Orbit	Avg.	Deg.	Clus.	Orbit	Avg.
DeepGMG	0.220	0.950	0.400	0.523	-	-	-	-	-	-	-	-
GraphRNN	0.080	0.120	0.040	0.080	0.017	0.062	0.046	0.042	0.064	0.043	0.021	0.043
GraphAF	0.18	0.20	0.02	0.133	1.669	1.283	0.266	1.073	-	-	-	-
GraphDF	0.06	0.12	0.03	0.070	1.503	1.061	0.202	0.922	-	-	-	-

Appendix

Table 12: Additional experiments on generic graph datasets.

Method	QM9			ZINC250k
Method	Val. w/o corr. (%) $\uparrow$	NSPDK MMD $\downarrow$	FCD $\downarrow$	Val. w/o corr. (%) $\uparrow$	NSPDK MMD $\downarrow$	FCD $\downarrow$
GraphAF	67.00	0.020	5.268	68.00	0.044	16.289
GraphDF	82.67	0.063	10.816	89.03	0.176	34.202
MoFlow	91.36	0.017	4.467	63.11	0.046	20.931
EDP-GNN	47.52	0.005	2.680	82.97	0.049	16.737
GraphEBM	8.22	0.030	6.143	5.29	0.212	35.471

Appendix

Table 13: Additional experiments on QM9 and ZINC250k datasets.

Table 16

Table 17

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