Alignment Matters Inside and Out in Equivariant Graph Flow Matching

Moritz Piening; Christian Wald

cs.LGAug 27, 2026

Alignment Matters Inside and Out in Equivariant Graph Flow Matching

Authors: Moritz Piening, Christian Wald

Organizations: Institute of Computer Science University of Göttingen Göttingen, Germany · Institut Camille Jordan INSA Lyon Villeurbanne, France

Abstract

Graphs are invariant under node permutations, motivating permutation-equivariant architectures in generative models. In flow matching, however, symmetry also affects the source-target coupling: one must choose both which graphs to pair and which node representatives to align. We study these two forms of alignment, termed outer and inner alignment, and their effect on equivariant graph flow matching. We connect inner alignment to transport on the graph quotient space, whose Euclidean quotient metric coincides with Gromov-Monge distance, and show that quotient couplings admit aligned representative lifts while symmetrization yields equivariant flow-matching minimizers, including for categorical endpoints. In practice, we compare random augmentation, permutation-blind minibatch optimal transport, approximate Gromov-Wasserstein alignment, and combinations of inner and outer alignment. Across continuous graph and molecular generation, alignment can substantially simplify trajectories and improve few-step generation, while its benefits depend on the alignment and computational budget. Our results highlight that effective graph flow matching benefits from alignment both inside and across graph pairs.

Figures & tables

Figure 1: Cycle-graph trajectories (

t=0

: bottom,

t=0.5

: center,

t=1

: top) . Colours mark source, lines node paths, and edge width edge value. At

t=0.5

, dense graphs become sparse circles as augmentations

\mathcal{A}

grow towards GW (top). Outer coupling straightens paths (bottom).

Method	Random	MBOT 4	MBOT 64	MBOT 1024	BoA 4	BoA 64	BoA 1024	GW	BoA 4	BoA 64	BoA 1024	GW
									$+$ BoA ${}^{4}_{\mathrm{out}}$	$+$ BoA ${}^{64}_{\mathrm{out}}$	$+$ BoA ${}^{1024}_{\mathrm{out}}$	$+$ GW ${}^{16}_{\mathrm{out}}$
Roundness ( $\downarrow$ )	$12.9_{8.9}$	$18.9_{4.9}$	$8.8_{4.0}$	$7.1_{3.0}$	$26.1_{7.0}$	$16.4_{7.2}$	$7.9_{3.2}$	$0.7_{1.0}$	$11.4_{3.0}$	$5.9_{1.2}$	$5.5_{2.0}$	$\mathbf{0.0}_{0.0}$
Degree ( $\to 2$ )	$1.90_{0.28}$	$2.15_{0.09}$	$1.81_{0.14}$	$1.80_{0.12}$	$1.91_{0.09}$	$1.93_{0.13}$	$1.94_{0.03}$	$\mathbf{2.00}_{0.00}$	$1.96_{0.08}$	$1.87_{0.02}$	$1.97_{0.01}$	$\mathbf{2.00}_{0.00}$

Table 1: Cycle-graphs: Roundness and mean degree are reported as mean std . Best is bold.

Figure 2: Mean aligned distance vs. CPU time per pair. Labels give

(\mathcal{A},B)

. Dotted: extrapolated least-squares

\mathcal{A}B

power-law fits (distance:

c_{\infty}+a(\mathcal{A}B)^{-p}

, time:

t_{0}+b(\mathcal{A}B)^{q}

Table 2: Molecular short-budget sweep: ( 2(a) ) Flow ODE at

5/25/125

Euler steps and ( 2(b) ) flow map at

1/2/5

steps. Rows are mean

{}_{\text{std}}

over

3

evaluation repeats. Best (

\pm 2\sigma

) is bold.

	QM9		ZINC250k
Method	Validity ( $\uparrow$ )	FCD ( $\downarrow$ )	Validity ( $\uparrow$ )	FCD ( $\downarrow$ )
GraphAF ( Shi et al., 2020 )	$67.14$	$5.246$	$67.92$	$16.128$
MoFlow ( Zang and Wang, 2020 )	$92.03$	$4.536$	$63.76$	$20.875$
GDSS ( Jo et al., 2022 )	$95.72$	$2.900$	$97.01$	$14.656$
CatFlow ( Eijkelboom et al., 2024 )	$99.81$	$0.441$	$99.21$	$13.211$
DiGress ( Vignac et al., 2023 )	$98.29$	$0.095$	$94.98$	$3.482$

Table 3: Molecular full-budget run. Ours:

500

Euler steps,

10{,}000

samples, mean std over three repeats. Baselines follow published protocols. GraphAF, MoFlow, and DiGress values are quoted from ( Hou et al., 2026 ) , DeFoG values from ( Xiong et al., 2026 ) .

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

	FGW–NNA ( $\to 0.5$ )			Degree MMD ( $\downarrow$ )			Clustering MMD ( $\downarrow$ )			Graphlet-orbit MMD ( $\downarrow$ )
Method	$5$	$25$	$125$	$5$	$25$	$125$	$5$	$25$	$125$	$5$	$25$	$125$
Random	$0.796_{.012}$	$0.581_{.021}$	$0.561_{.015}$	$0.114_{.045}$	$0.067_{.016}$	$0.059_{.011}$	$0.179_{.100}$	$0.085_{.042}$	$0.075_{.028}$	$0.062_{.033}$	$0.046_{.033}$	$0.043_{.025}$
MBOT 16	$0.687_{.018}$	$0.561_{.016}$	$0.530_{.007}$	$0.031_{.004}$	$0.023_{.006}$	$0.015_{.005}$	$0.090_{.014}$	$0.046_{.010}$	$0.037_{.010}$	$0.006_{.001}$	$0.011_{.001}$	$0.007_{.000}$
FLB	$0.725_{.013}$	$0.558_{.011}$	$0.555_{.022}$	$0.054_{.004}$	$0.037_{.011}$	$0.030_{.010}$	$0.109_{.037}$	$0.042_{.006}$	$0.040_{.011}$	$0.030_{.014}$	$0.027_{.017}$	$0.021_{.015}$
FLB $+$ FLB ${}^{8}_{\mathrm{out}}$	$0.691_{.040}$	$0.562_{.019}$	$0.547_{.009}$	$0.034_{.011}$	$\mathbf{0.012}_{.002}$	$0.013_{.007}$	$0.121_{.041}$	$0.045_{.024}$	$0.047_{.021}$	$0.018_{.011}$	$\mathbf{0.007}_{.002}$	$0.010_{.005}$
GW	$0.577_{.011}$	$\mathbf{0.522}_{.004}$	$0.520_{.023}$	$0.027_{.013}$	$0.014_{.002}$	$0.012_{.004}$	$0.049_{.019}$	$\mathbf{0.015}_{.005}$	$0.016_{.007}$	$0.014_{.004}$	$0.010_{.003}$	$0.008_{.001}$

Appendix

Table 4: Continuous SBM, unconditional mixture over

K\in\{1,\dots,5\}

(

N=10

), at

5/25/125

Euler steps. Rows are mean

{}_{\text{std}}

over

3

training seeds (

5

evaluation repeats each). Best value per column is bold (MMDs (degree, clustering, graphlet orbit)

\downarrow

, Best FGW–NNA:

\sim 0.5

	FGW–NNA ( $\to 0.5$ )			Degree MMD ( $\downarrow$ )			Clustering MMD ( $\downarrow$ )			Graphlet-orbit MMD ( $\downarrow$ )
Method	$5$	$25$	$125$	$5$	$25$	$125$	$5$	$25$	$125$	$5$	$25$	$125$
Random	$0.746_{.009}$	$0.580_{.008}$	$0.544_{.021}$	$0.043_{.009}$	$0.008_{.002}$	$0.005_{.001}$	$0.076_{.010}$	$0.029_{.007}$	$0.020_{.004}$	$0.028_{.002}$	$0.006_{.004}$	$0.003_{.003}$
FLB	$0.714_{.010}$	$0.543_{.017}$	$0.558_{.023}$	$0.023_{.004}$	$0.005_{.001}$	$0.005_{.001}$	$0.047_{.006}$	$0.021_{.003}$	$0.017_{.003}$	$0.018_{.003}$	$0.005_{.001}$	$0.003_{.002}$
FLB $+$ FLB ${}^{8}_{\mathrm{out}}$	$0.699_{.007}$	$0.552_{.018}$	$0.539_{.007}$	$0.013_{.002}$	$0.007_{.002}$	$0.007_{.001}$	$0.049_{.009}$	$0.030_{.005}$	$0.027_{.010}$	$\mathbf{0.009}_{.006}$	$0.005_{.003}$	$0.005_{.004}$
GW	$0.584_{.002}$	$\mathbf{0.523}_{.010}$	$\mathbf{0.524}_{.003}$	$0.011_{.001}$	$\mathbf{0.005}_{.001}$	$\mathbf{0.004}_{.001}$	$\mathbf{0.016}_{.002}$	$\mathbf{0.011}_{.001}$	$\mathbf{0.010}_{.001}$	$0.016_{.001}$	$0.005_{.002}$	$0.003_{.001}$
GW $+$ GW ${}^{8}_{\mathrm{out}}$	$\mathbf{0.561}_{.017}$	$0.526_{.028}$	$0.529_{.014}$	$\mathbf{0.008}_{.001}$	$0.005_{.001}$	$0.005_{.002}$	$0.018_{.001}$	$0.014_{.001}$	$0.014_{.004}$	$0.012_{.001}$	$\mathbf{0.004}_{.001}$	$\mathbf{0.003}_{.001}$

Appendix

Table 5: Continuous SBM, class-conditional on

K\in\{1,\dots,5\}

(

N=10

), evaluated by pooling generated and real graphs with the same ground-truth

K

multiset. Mean

\pm

std over

3

training seeds (

5

evaluation repeats each) at

5/25/125

Euler steps. Descriptor MMDs: lower is better, FGW–NNA: closest to

0.5

is better. Best per column in bold.

Setting	Validity	FCD
Base + RRWP	$0.8846\pm 0.0030$	$12.3371\pm 0.1229$
+ latent noise	$0.9080\pm 0.0012$	$11.2470\pm 0.0704$
+ self-conditioning	$0.9307\pm 0.0010$	$8.8572\pm 0.0763$
+ edge weight $5$	$0.9606\pm 0.0019$	$6.9861\pm 0.0231$

Appendix

Table 6: Cumulative ZINC250k ablation at

25

Euler steps and reduced training budget. Higher validity is better and lower FCD is better.

	Cycle graphs	Continuous SBM	QM9	ZINC250k
Channels $C$	$3$	$2$	$11$	$16$
Nodes $N$	$12$	$10$	$\leq 9$	$\leq 38$
Gradient batch ( $\neq B$ )	$16$	$16$	$64$	$32$
Epochs	$\approx 819$ ( $204{,}800$ steps)	$1000$	$100$	$100$
Time embedding dim	$32$	$32$	$64$	$64$
Head	velocity (MSE)	velocity (MSE)	endpoint (cross-entropy)	endpoint (cross-entropy)

Appendix

Table 7: Per-experiment settings for the controlled coupling comparisons, shared across all coupling conditions. Architecture, optimizer, EMA, gradient clipping, and learning-rate schedule are identical throughout (see text), where only the entries below vary. The larger-model protocol is specified separately in Appendix E . “Head” is the flow-matching parametrization: MSE on the velocity field, or cross-entropy on the predicted categorical endpoint.

Panel of Figure 3 — Figure 3: Validity during training at $125$ Euler steps, averaged over $5$ repeats of $1024$ molecules. The shaded bands show $\pm 1$ standard deviation.

Table 8: Molecular short-budget sweep, uniqueness (same runs and evaluations as Table 2 ): ( 8(a) ) Flow ODE at

5/25/125

Euler steps and ( 8(b) ) flow map at

1/2/5

steps. Rows are mean

{}_{\text{std}}

over

3

evaluation repeats. Best is bold (up to

\pm 2\sigma

of evaluation variability).

	Continuous SBM		QM9		ZINC250k
	$N=10$		$N=9$		$N=23$
Method	ms/pair	Corr. to GW	ms/pair	Corr. to GW	ms/pair	Corr. to GW
GW	$1.30$	—	$2.21$	—	$2.66$	—
FLB	$0.053$	$0.64$ ( $0.56$ )	$0.061$	$-0.38$ ( $-0.34$ )	$0.065$	$-0.45$ ( $-0.46$ )
Euclidean	$0.004$	$0.42$ ( $0.38$ )	$0.004$	$0.77$ ( $0.77$ )	$0.006$	$0.90$ ( $0.89$ )
BoA 32	$0.036$	$0.72$ ( $0.67$ )	$0.043$	$0.91$ ( $0.91$ )	$0.16$	$0.95$ ( $0.94$ )

Appendix

Table 9: Runtime in ms per pair (CPU, single thread) and correlation with the GW cost, Pearson (Spearman), over 300 pairs of a training source sample and a same-

N

training graph. Euclidean is the MBOT cost

\|E-F\|^{2}

and BoA A its minimum over

\mathcal{A}

random relabellings.

	Continuous SBM	QM9	ZINC250k
	$N=10$ , $B=16$ , $\mathcal{A}=32$	$N=9$ , $B=64$ , $\mathcal{A}=32$	$N=23$ , $B=32$ , $\mathcal{A}=32$
MBOT B	$0.30$	$1.74$	$1.71$
FLB	$0.75$	$3.39$	$1.94$
FLB $+$ FLB ${}^{8}_{\mathrm{out}}$	$1.48$	$6.54$	$3.68$
BoA A	$0.51$	$2.62$	$5.07$
BoA ${}^{\mathcal{A}}+$ BoA ${}^{B}_{\mathrm{out}}$	$0.82$	$9.82$	$31.14$

Appendix

Table 10: Alignment cost per training batch in ms (CPU, single thread). Outer FLB and GW reorder in sub-batches of eight. MBOT and BoA out assign over the full minibatch.

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Equivariance is central to graph generative models, as it ensures the model respects the permutation symmetry of graphs. However, strict equivariance can increase computational cost due to added architectural constraints, and can slow down convergence because the model must be consistent across a large space of possible node permutations. We study this trade-off for graph generative models. Specifically, we start from an equivariant discrete flow-matching model, and relax its equivariance during training via a controllable symmetry modulation scheme based on sinusoidal positional encodings and node permutations. Experiments first show that symmetry-breaking can accelerate early training by providing an easier learning signal, but at the expense of encouraging shortcut solutions that can cause overfitting, where the model repeatedly generates graphs that are duplicates of the training set. On the contrary, properly modulating the symmetry signal can delay overfitting while accelerating convergence, allowing the model to reach stronger performance with

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