GGFlow: A Graph Flow Matching Method with Efficient Optimal Transport
Authors: Xiaoyang Hou
Tian Zhu
Milong Ren
Dongbo Bu
Xin Gao
Chunming Zhang
Shiwei Sun
Venue: Accepted by TMLR ()

Abstract: Generating graph-structured data is crucial in various domains but remains challenging due to the complex interdependencies between nodes and edges. While diffusion models have demonstrated their superior generative capabilities, they often suffer from unstable training and inefficient sampling. To enhance generation performance and training stability, we propose GGFlow, a discrete flow matching generative model incorporating an efficient optimal transport for graph structures and it incorporates an edge-augmented graph transformer to enable direct communications among edges. Additionally, GGFlow introduces a novel goal-guided generation framework to control the generative trajectory of our model towards desired properties. GGFlow demonstrates superior performance on both unconditional and conditional generation tasks, outperforming existing baselines and underscoring its effectiveness and potential for wider application.

Published in Transactions on Machine Learning Research (12/2025)
GGFlow: A Graph Flow Matching Method with Efficient Optimal Transport
Xiaoyang Hou"
houzkhoyoog2222icc..o.om
Institute of Computing Technology University of the Chinese Academy of Sciences Tian Zhu" Institate of Computing Technology
shution222@ict..c.com
University of the Chinese Academy of Sciences Milong Ren
rexmilong@1bBiictoc.cm
Institute of Computing Technology
University of the Chinese Academy of Sciences Dongbo Bu
dbwWicl.ac.cm
Institate of Computing Technology
University of the Chinese Academy of Sciences
Central China Research Institute for Artificial Intelligence Technologies Chunming Zhang
shanghhunminng@niii..c.oo
Institute of Computing Technology
Western Institate of Computing Technology
Phil Rivers Technology
Xin Gao
xin. Sin.gao@kaast..ed edu.
King Abdullah University of Science and Technology Shiwei Sun
dwsum Dict oc.
Institute of Computing Technology
University of the Chinese Academy of Sciences
Western Institute Computing Technology
Reviewed on OpenReview: /ttp::///penrevviee formm? id= K8RLXXtM0000
Abstract
Generating graph-structured data is crucial in various domains but remains challenging due to the complex interdependencies between nodes and edges. To enhance generation performance, propose GGFlow, a discrete flow matching generative model with an efficient optimal transport for graph structures and it incorporates edge-augmented graph transformer. Additionally, GGFlow introduces novel goal-guided generation framework to control the generative trajectory of our model towards desired properties. GGFlow demonstrates superior performance on both unconditional and conditional generation tasks, outperforming existing baselines and underscoring its effectiveness and potential for wider application. Code: bicts://githut www./oo///aaaeeeee Coom//iaaoyag338//FF1oo
1 Introduction
Graph structural data generation has become critically important across various domains, including social networks Grooer et al. 2019), drug design (Bilodeau et al. 2022), and neural architecture search (NAS) (Lee et al. 2020). Effective modeling of the intrinsic joint distribution and accurate description of topological structures graphs are essential for these applications. Deep generative models have increasingly demonstrated success in graph generation by effectively modeling the complex structural propertics of graphs. These models are typically categorized into autoregressive and one-shot types. Autoregressive models, such as GraphRNN
"Contributed equally: Author order is randomized and adjusted needed for individual purposes. "Corresonneecee should be addressed Shiwei Sun (dwsunkict.ac.co and Xin Gao (xin.gao&kkasst.oom Sin.gao@kauss..chhs.

---

Published in Transactions on Machine Learning Research (12/2025)
You et al. components. In contrast, one-shot methods generate entire graphs in a single step, more effectively capturing
2018). generate graphs sequentially, often overlooking the interdependencies among all graph
the joint distribution (Ma et al. 2018 Luo et al. 2023 Niu et al. 2020
Diffusion models have shown great promise and achieved significant performance in various domains Ho et al. 2020 Song et al., 2020 Ho et al., 2022). In the context of graph generation, diffusion models have been adopted to enhance generative capacity. EDP-GNN and GDSS are among the first to utilize diffussin models for graph generation, adding continuous Gaussian noise to adjacency matrices and node types, which may lead to invalid graph structures (Niu et al., 2020 Jo et al. 2022b Due to the inherent sparsity and discreteness of graph structures, GSDM enhances model fidelity by introducing Gaussian noise within a continuous spectrum space of the graph, DiGress and PPGN apply discrete diffusion models for graphs (Luo
et al. 2023 Vignac et al.) 2022 Austin et al. 2021 Haefeli et al. 2022 Huang et al.) 2023
Flow matching generative models also offer a compelling altornative. As these methods transform the training paradigm from SDEs to more stable interpolation method, they enable more efficient sampling via ordinary differential equations (ODEs) (Lipman et al. 2022, Song et al. 2024 Yim et al., 2023 Furthermore, the application of optimal transport (OT) has been explored to straighten marginal probability paths, leading to reduced training variance and accelcrated sampling (Bose et al.
2023
Tong et al.) 2023 Klein et al.
2024; Pooladian et al. 2023). Nevertheless, the integration of OT into graph-based systems often encounters substantial computational hurdles due to the inherent complexity of computing OT metrics for discrete structures (Chen et al. 2020b Petric Maretic et al. 2019).
In this paper, we untroduce GGFlow, novel generative model that leverages discrete flow matching techniques with an efficient optimal transport coupling in graph generation. The model preserves graph sparsity and permulation invariance, which is essential for realistic graph generation. Additionally, GGFlow employs a goal-guided framework using reinforcement learning for conditional generation. GGFlow achieves state-of of-the- art results in both unconditional and conditional graph and molecule generation tasks and surpasses existing methods. Its effectiveness in conditional generation tasks underscores the practical impact of our approach.
Our contribution can be summarized as:
GGFlow introduces the first discrete flow matching generative model with an efficient optimal transport for graph data, and it also incorporates an odgo-angmented graph transformer to enhance generation lasks further.
GGFlow proposes novel guidance framework using reinforcement learning to control probability flow during graph generation, targeting specific properties.
GGFlow demonstrates state-of-the-art performance in various unconditional and conditional graph generation tasks, consistently outperforming existing mothods across diverse graph types and com- plexities.
2 Related Work
2.1
Flow Matching and Diffusion Models
Diffusion models have gained widespread popularity various fields, including computer vision, natural language processing, and biological sciences, demonstrating notable success in generative tasks (Ho al.
2020 Song et al.] 2020 Watson et al.] 2023 Ingraham et al.] 2023 Liu et al. 2024a Ren al. 2024 Zhu et al. 2024] Recently, flow matching generative models have emerged more efficient and stable alternative (dctails in Appendix A.1]) improving generative performance (Lipman et al. 2022 Song et al.
2024 Campbell et al.] 2024). Some approaches further enhance performance by incorporating optimal
transport. The generative processes of these models are summarized in Figure 1
Previous works (Campbell et al., 2024: Gat et al. 2024) extended flow matching to discrete spaces, while Eijkelboom et al. (2024) applied variational flow matching graphs, but without adequately addressing key graph-specific properties such adjacency matrix sparsity. GGFlow tackles these challenges by introducing
2

---

Published in Transactions on Machine Learning Research (12/2025)
discrete flow matching model with efficient optimal transport tailored for graph data. Furthermore, propose novel framework for guiding the generative process, enhancing its practical applicability.
Diffusion
Flow Matching
Flow Matching
Optimal Transport
Figure Illustration of generative trajectories using different methods. The generative trajectories are learned by the diffusion model (left), flow matching model (center), and flow matching model with optimal transport (right).
2.2 Graph Generative Models
Graph generative models are typically categorized into two main types: autoregressive and one-shot models. Autoregressive models generate graphs soquentially (Wang ct al. 2018 You et al.) 2018 Jin al.) 2018 Shi eet al. 2019 Luo et al. 2021 Kong al. 2023). While effective, these models often computationally expensive and fail to account for permutation invariance, a crucial property for graph data, resulting in potential inefficiencies. In contrast, onc-shot models aim to capture the distribution of all graph components
simultancously (De Cao & Kipf 2018 Ma et al. 2018 Zang & Wang 2020), better reflecting the inherent
interactions within graphs. Despite the advantages, diffusion-based one-shot models (Niu et al. 2020 Jo et al.
2022b Vignac et al. 2022
Chen et al. 2023 Bergmeister et al.) 2023; Luo et al.) 2023 Haefeli et al. 2022
Yan al. 2023 Jang et a..
2023 Madeira et al.) 2024 Bergmeister al. 2024 Chen et al. 2023) Minello
et al. 2024 Zhao et al. 2024
Xu et. al. 2024) show promising results in downstream tasks. GGFlow employs
discrete flow-matching generative model, achieving superior generative performance. More comparisons
with recent works are presented in Appendix B
3 Background
Discrete Flow Matching (DFM) generative models extend the flow matching paradigm to generate data samples within a discrete domain. We consider a discrete variable v, taking values a finite state space dcnoted by N -1,....,N; The marginal distribution of Ia represented by the probability vector pt, satisfying Σ (I = i) = 1. The process interpolates between a predefined noise prior Po = Prof (typically uniform distribution or mask distribution) and the target data distribution P1 = Pdata (Campbell et al.
2022
Gat et al. 2024). The marginal probability path is defined as:
(x(x)=
Σ pr(z|21, (0), Bo.TiEN
(1)
where the training pairs are sampled from the joint distribution T(21, In the simplest scenario, is modeled as an independent coupling, i.e., (((1,=0) (x1
Following Campbell et al. (2024), the conditional probability path Pij Pqj((,,/a (x,/2), 20) defined via linear interpolation. Given data sample 21 and a prior sample zo, the path and independent coupling are given by:
Pin [ Ho) = tō(1,, x1) + (1 t)6(2,, T(I = Paata (2)))(()))))))
(2)
(3)
where 8 denotes the Kronecker delta function.
3

---

Published in Transactions on Machine Learning Research (12/2025)
For the sampling procedure, DFM leverages Continuous-Time Markov Chain (CTMC) formulation. The CTMC is characterized by an initial distribution po and a time-dependent probability rate matrix (analogous to velocity field) ux(+,-) E RNXN, The transition dynamics over a small step At are defined as:
(e 6(11+A1, +2( Att
(4)
where (z (++2)) represents the transition rate between states. Intuitively, (t+ ar) yields the probability that transition from state Iy to state T24 will occur in the next infinitesimal time step At. To ensure proper normalization such that Extts. (1 as ue (1t, = Exez, u4 (1 k), while off-diagonal entries are non-negative. Consoquently, the marginal distribution evolves according the Kolmogorov forward equation:
= 1, the diagonal entries are defined
d.p: = un Pt
Specifically, DFM relies on conditional rate matrix (21) E BNXN to govern the denoising process. Under mild assumptions, [Campbell et al. (2024) derive a closed-form solution for this valid conditional rate matrix. For off-diagonal entries (I: # x1+))), this is defined as:
(Te, (44 = ReI.U[0:Pe (P+4a/))))))))))))))))))) (pp.n1 Zipan (2)(01)
(6)
where Z, 1(x Pen((a 0}| is a normalization constant. The marginal velocity field is then obtained (x. (1111) which can be substituted into Equation (4) for
via expectations, (e generation.
The discrete flow matching formulation introduced by Campbell et al. 2024) was recently adapted for discrete graph generation in DeFoG (Qin el al. 2024). DeFoG leverages discrete flow matching with independent coupling to model the generation of each node and edge independently, successfully demonstrating the benefits training-samplee Optimal Transport (OT) coupling into this discrete graph flow matching framework -ramewrk--soceffiaall concerning its impact sample quality, and the smoothness of the learned rem-remains critical, underexplored avenue. To address this critical gap and fully harness the theoretical advantages of OT, introduce method, GGFlow.
disentanglement in the graph domain. However, the utility of incorporating
4 Methods
In this section, we present our methodology, GGFlow. Section 4.1 outlines the discrete flow matching method for graph generation. Section 4.2 covers an efficient optimal transport for graph flow matching. Section
introduces GraphEvo, neural network for graph generation. Section 44 examines the permutation
properties of GGFlow, and Section 4.5 discusses goal-guided graph generation using reinforcement learning.
4.1 Discrete Flow Matching for Graph Generation
A graph G = EV,F), where V and E denote the sets of nodes and edges, has a distribution denoted by PG) (p (E)). The attribute spaces for nodes and edges are and E, with cardinalities and m, respectively. The attributes of node i and edge ij are denoted by es E V and eij E, the node and edge probability mass functions (PMF) are p(u = a) and p(en) = b) where a E {1...., and b mL..
We formulate the noising and generation trajectory by treating the attributes of each node and edge as independent random variables. Let x be a generic variable x E VUE. Under the assumption of independence across all components, the distribution of the noisy graph G at time t is defined as the product of the individual component probabilities:
(c A (oV (V'), (F
II p(c II p(xt). 1.k
(7)
REVUE
4

---

Published in Transactions on Machine Learning Research (12/2025)
where G denotes the set of all node and edge variables, and pc(x') is the PMF of single variable at time
The node and edge embeddings in the graph are given by matrices V E R Nxd, and E E R respectively, where N is the number of nodes, d, is the hidden dim of node embeddings and de is the hidden dim of edge embeddings. We denote the transpose of matrix A as A and A' represents the state of matrix A at time We use discrete flow matching to model the graph generation process.
Source and target distribution GGFlow aims transform prior distribution GD Pref to target data distribution G¹ Pdata- The training data (G", G¹) are sampled from joint distribution TT G)), satisfying the marginals constraints Pref = Eos TTG², G²).. Pdata = -Go
To account for graph sparsity, the prior distribution Prof - (D Pfot) is designed to approximate the true data distribution closely. To ensure the permutation invariance of the model, the priors are structured as products of single distributions for all nodes and edges: II. 1/2 x II Ctj (Vignac et al. 2022). Further details on the prior can be found in Appendix C11
Probability path and coupling We define probability path p. (G') that interpolates between source distribution Pref and target distribution Pdata ie. po Prof and pt Pdata The marginal probability path is given by:
Pr(G')=
Σ
(G)
(8)
(GT
where
Guj(G G*) +
(6(v.v) + (1 --
(6((,,')) + (11_))(zz==x
SEVUE
8 is the Kronecker delta, indicating equality of the indices, the sampling procedure is independent for cach node and edge. Given the sparsity of both the prior and data distributions, we can infer that the intermediate distribution similarly sparse, aiding model training.
Critically, we employ the Optimal Transport (OT) map as the coupling between the marginals. Compared the independent coupling used in previous works (Campbell al.) 2024 Qin al.) (024), OT provides generalized and more structured coupling that enhances the quality of conditional samples (Tong al. 2023
Gat et al. 2024) (detailed in Section 4.2
We define probability velocity field:
GG a V), u
E)) & II ux(x" .x)
aEVUE
for GGFlow, which generates the probability path from Equation
The probability velocity field u,(G", .G)
is derived from the conditional probability velocity field w/ G o Gt), and can be expressed as:
u G)
Σ u. GIG", P. G)) (G)
(9)
G G2) L PI (TT GT)
GG = pi (G G Go)
=
(10)
GGFlow chooses the conditional marginal probability (G" GIGo, G²) as:
_f14, x0)
MG(G',G|' G!) A II
ux(x,, x/1= x II
Z - (1 + 5f 6f2, =1x,,111)
(11)
ZEVUE
ZEVUE
5

---

Published in Transactions on Machine Learning Research (12/2025)
where Z, = = (e(aa 2 0, x: E EUE More details about the choice of the conditional vector field are provided in Appendix C.2
Training objective Given the intractability of the posterior distribution Pit (G' (G G" ,"), we approximate
(
it as Pije (G",")) ,G) using neural network, GraphEvo, as detailed in Section 4.3 The training objective is formulated as:
L = Eupca (G ((..)(())))))(( (poni""))))))(((((())) llg [G
(12)
where 1) is uniform distribution on [0, I].
Sampling Procedure
In the absence of the data distribution G during sampling, we simplify the generative process (G"+A (G" without the calculation of the full expectation over the conditional
vector field ( "", GO) G.)):
+ PHAAI (G G = [p ,G)) + uz G G.GG)AAt)
Σ P G',G', G¹ Gi) GO).
(13)
This expectation can be sampled efficiently using two-step procedure (Algorithm D (1) first, sample the endpoint & ~ P1je (C GO G9) using the approximale posterior, and (2) then, we sample the next state G++ conditioned on G¹. The conditional sampling transition given by:
Gi+ ~ P (G+ G G, G)) 5(G ,G' + ue G G ())
where the term (G", GT At represents the conditional probability of a transition from Gt to G++AI in the infinitesimal time step At, given the initial condition GD and the sampled endpoint G1. Further details on the sampling and training procedures are provided in Algorithms and 3
Algorithm Sampling Procedure of GGFlow
Require: 1 0, (pV rof) (C',, GICo, G'), Nateps
1: At = 1/Nstops
for n E (0...., Nateps do P .G) GraphiEv(GG'
3:
4: G¹ " G", )o)
// Sampling from the conditional velocity field
6:
at+Ae Pt+ ( G², C")
6(C G²) u. (CtAyc")
1=1+ Al
8: end for
9: return
G G² (VL, F
4.2 Efficient Optimal transport for graph flow matching
Optimal transport (OT) has been effectively applied to flow matching generative models in continuous
variable spaces, improve generative performance (Tong et al.) 2023 Bose al. 2023 Song et al. 2024). To generalize this for graphs, we set the joint distribution TT, G' as a coupling to the 2- Wasserstein OT map, which minimizes the 2- 2asssersteiin distance between Pref and Pdata- Lel G denote the space of graphs. The optimal transport plan x² (Go, G²) is defined as the minimizer of the expected cost (Bookstein et al.
2002
TTG G²) arg inf / nEP(po.p:)
H(G (IG",G")dT(G") G+),
(14)
where
H = 500 (00.. + Σ e(j))
(15)

---

Published in Transactions on Machine Learning Research (12/2025)
where (ppo. P1) represents the set of all joint probability measures (couplings) on 9 x 9 whose marginals are Po and pl, respectively, where G = (VK (uf ), F* Jty), K = 0,
The practical application of OT to large datasets is computationally intensive, often requiring cubic time complexity and quadratic memory [Togg et al. minibatch approximation of OT (Fatras al.
Villani 2009 To address these challenges, we use a 2021 A detailed analysis of optimal transport during the
2020
training procedure is provided in Appendix E
4.3 GraphEvo: Edge-augmented Graph Transformer
Our neural network, Graphlivo, is tasked with predicting the necessary factorized posterior distribution, PiI[ G"). Given the intermediate graph state G = [V', E'} and the time step GraphEvo oulputs the predicted distribution for the final state G
pnp P =
( (Vi Pipe (E (F
For theoretical consistency with the flow matching framework, the full posterior is denoted as Dije(G"IG G acknowledging the implicit dependency on the initial noise state G via the timo-dependent path. The model itself, however, primarily operates on Gt and l.
Since structural information and edge relations
critical node attributes for capturing complex graph topologics and are known to enhance link generation tasks (Hussain et al. 2024 Hou et al. 2024
Jumper et al. 2021), GraphEvo is built upon specialized Transformer architecture. To effectively capture non-local dependencies and higher-order structural motifs, we extend the transformer with a triangle attention mechanism for dynamic edge updates. Furthermore, we augment the input features of the graph, represented by node and edge embedding matrices V R-xxd, and E N (where N the number of nodes, dy is the hidden dim of node embeddings and de is the hidden dim of edge embeddings.), with rich global auxiliary graph features y E Rixh, such cycles and the number of connected components (Vignac al. 2022). Here, h is the dimension of the auxiliary features. This comprehensive approach enables GraphEvo to efficiently and accurately model the joint distribution of all graph components, as detailed in Algorithm 2
4.4 Permutation Property Analysis
Graphs are invariant to random node permutations, and GGFlow preserves this property. To ensure permutation invariance, we analyze the permutation properties of our neural network, training objectives, and conditional probabilities path. First, we analyze the permutation invariance of the training objectives (Vignac et al. 2022). Since the source and target distributions permutation invariant, the independent coupling also exhibits this invariance. Our optimal transport map, derived from Equation [4 similarly demonstrates invariance to identical permutations. Further clarifications regarding optimal transport can be found in Appendix C.4
Theorem If the distributions p(G) and p(G²) are permutation invariant, and the cost function maintains invoriance under identical permutations, i.e., H(G",G*) H oG*) for any permutation o, then the oplimal bransport map æ also exlubiis invrrinnce under identical permulations, such that. T( G+) (G))= TT (oG oG"))
Proof of this theorem can be found in Appendix C.4 To ensure that the generated graph retains its identity under random permutations, the generated distribution must remain exchangeable, and GraphEvo must be permutation cquivariant.
Proposition 1.
The distribution generated by the conditional flow is exchangeable with respect to nodes and Fa)), (p"(o (o'V),pB(o" (o)) where is a permulation operator.
grophs, iec. p(G) (p" Proposition 2. GraphEvo is permutation equivariant.
The proofs of Proposition and are provided in Appendix C33 and Appendix
D.1 respectively.

---

Published in Transactions on Machine Learning Research (12/2025)
Algorithm 2 Architecture of GraphEvo
Require: G¹, N,,te
1: // Initialization Phase
2: Vt, E² G G
3: y < ExtractFeaturo(("".
TimcEmbedding(()
y < y
V.E.y <- Linear(V"), Lincar( (F*), Lincar()
// Triangle Attention Phase
for 0, 0,1....,,NAa. do
8:
// Definition: Triangle-attention Mechanism Computation
// 1. Node Attention
10:
Qv, Kv, V L.innner(()))
Yv+ 9.xk5
11:
vdx
12:
1/2. Incorporate edge features to celf-attention scores Y 7.+Fill +FFMM.( E)
13:
14:
Etmp Y
15:
//3. 3. Edge Triangle Attention
16:
Qe, Ke, V b. +- Linear(Eunn) (Emm))
Yo + 9o +b
17:
Vde
18:
E' Ye V. sigmoid(g)
19:
E' Linear FiLM(EE,,)) // Incorporate global to edge
20:
// 4, Update Node and Global features
21:
V Y, * V.
V Linear FiLM(V', .y) // Incorporate global node
22:
23:
y + Linear Linear(y) PNA(VV + PNA(EE)
24:
// Food-Forward & Residual Connections
25:
V + ReLU
LayerNorm(V (ropouu(V'))
26: E ReLU
LayerNorm E Dropout(E'))
27:
ReLU LayerNorm(y Dropout
28: end for
29: // Prediction Phase
30: P. (V [tlll,ttvv")@ft v (E "EIEEEE),, E.),y + Linear(V), Lineerr))) Linear(y)
31: GG (C GO) = -pP (V'VV, (8),E,,(EE))) B²))
32: return Pip GG

---

Published in Transactions on Machine Learning Research (12/2025)
4.5 Goal- Guided Framework for Conditional Generation
For practical applicalions such as drug discovery, we propose goal-guided framework for discrete flow matching, employing reinforcement learning (RL) to guide graph flow matching models for non-differentiable objectives. The goal of the guidance method is to map the noise distribution Po to a preference data distribution pi using a reward function R('', t).
Formulation We cast the diffusion process as a Markov Decision Process (MDP). Concretely, our formulation builds upon prior work (Wallace al., 2024). We adopt and extend this trajectory-level RL formulation to discrete flow-mate now-matching. The state at decision epoch t is the flw-matching representation 8, = (G*, (G**,t) t), and the action space comprises all realizable successor states G++at at the next time t + At. P is the transition dynamics and policy network ((ar(st).
Reward function To align with practical scenarios where only the final outcome can be evaluated, we employ a sparse reward structure. The per-step reward function R(G', 1) is zero for all intermediate steps, and the final reward is defined solely on the generated sample G
R(G", 1) [G
(16)
where r(G") is the objective function (c.g., predicted molecular property). The RL agent seeks to maximize the total cumulative reward R = EiE(o.1) R(G", t), which simplifies to R = r(G").
Exploration To enable exploration, we additionally introduce a temperature parameter t for the policy network during sampling, allowing the model to explore a broader space at higher lemperatures:
(l 8.) (T =
= (bt, Gy) + mel: G')
(17)
As detailed in Section 4.1 this transition probability Pi+a (G*+ is efficiently sampled using two-step procedure: first sampling the & endpoint, and then sampling the next state G++ conditioned on the endpoint. Mcanwhile, we demonstrate in the 2-D Gaussian case that this approach is able to explore significantly larger region of the space (Appendix
Training The goal of RL training is to maximize the reward function. To prevent overfitting to the reward preference distribution, we add a Kullback-Leible (KL) divergence term between the Reinforcement learning fine-tuned model (-) and pre-trained model p-(-) @Ouyang et al. 2022).
We adapt the policy gradient method established in Wallace et al. 2024 to fine-tune our network parameters 0 for goal-guided generation. The policy p(ay)s:) is implicitly defined by the model's prediction of the final state p ( G.).
Similar to previous work (Liu et al. objective can be written as:
2024b under our definitions of reinforcement learning, the training
LRL= Pp R(( 1)
Σ log p RL (G Σ ( ("""" G") G"))]
B
(18)
where ( represents Pdata (G ( (G) G (C
Using this optimization objective, we fine-tune the pre-trained flow matching model to generate data following the preference distribution. By integrating optimal transport, optimize the pairing of prior data and high-reward training data. (Chen et al. 2020a). The pseudo-code for the guided GGFlow training is provided in Am 4 and toy example is shown in Appendix F
5 Experiments
To validate the performance of our method, we compare GGFlow with state-of-the-art graph generative basclines on generic graph generation and molecule generation, over several benchmarks in Section 5.1 and Section 5.2 respectively. The ability of GGFlow perform conditional generation is analyzed in Section 5.3 Finally, conduct detailed ablation studies presented in Section 5.4
9

---

Published in Transactions on Machine Learning Research (12/2025)
5.1 Generic Graph Generation
We evaluated GGFlow on five generic graph generation benchmarks of varying sizes: Ego-small, Community-
small, Grid, Planar and Enzymes. We employ the same train/test tain//testto split as GraphRNN ( (You et al. 2018),
utilizing 80% of each dataset for training and the remaining for testing. We compared GGFlow's performance
against. well-known autoregressive models: DeepGMG (Li et al. 2018), GraphRNN You et al. 2018)
GraphAF SSi ct al.
2019), and GraphDF (Luo et al. 2021) and one-shot models: GraphVAE. (Simonovsky
& Komodakis 2018).
GNF Liu et al. 2019). EDP-GNN (Niu et al. 2020), GDSS [Jo et al. 2022a), DiGress
[Vignac et al.) 2022).
GRASP (Minello et al. 2024, GSDM (Luo et al. 2023), GruM (Jo et al. 2024), and
SwinGNN (Yan et al.
2023). Consistent with previous studies, we generated an equal number of graphs
as the test set to compare distributions of graph statistics, including degree distribution (Deg.), clustering coefficient (Clus.); and the frequency of 4 node orbits (Orbit). Detailed descriptions of datasets, baselines, and metrics are provided in Appendix H
Table 1 presents our results, showing that GGFlow achieves superior performance across most metrics. Additionally, GGFlow demonstrates comparable performance compared to state-of-the-ard models in generating large graphs on the Grid dataset. These findings underscore the effectiveness of GGFlow at capturing the local characteristics and data distributions of graphs. Additional metrics and datasct experimental results are included in Appendix G and we visualize the generated graphs in Appendix J
Table 1: Generation results on the generic graph datasets. Results are the means of different runs. The best results and the second-best best results are marked bold and bold.
Ego-small
Community-small
Grid
Method
Step
Deg.
Chus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
Training Set
0.014
0.022
0.004
0.013
0.003
0.009
0.001
0.005
0.000
0.000
0.000
0.000
DeepGMG
0.040
0,100
0.020
0.053
0.220
0.950
0.400
0.523
GraphRNN
0.090
0.220
0.003
0.104
0.080
0.120
0.040
0.080
0.064
0.043
0.021
0.043
GraphAF
0.031
0.107
0.001
0.046
0.178
0.204
0.022
0.135
GraphDF
0.039
0.128
0.012
0.046
0.060
0.116
0.030
0.069
GNF
0.030
0.100
0.001
0.044
0.200
0.200
0.110
0.170
GraphVAE
0.137
0.166
0.051
0.118
0.358
0.969
0.551
0.626
1.594
0.000
0.904
0.833
EDP-GNN
0.054
0.092
0.007
0.051
0.050
0.159
0.027
0.079
0.460
0.243
0.316
0.340
1000
GDSS
0.027
0.033
0.008
0.022
0.044
0.098
0.009
0.058
0.133
0.009
0.123
0,088
1000
GSDM
0.020
0.050
0.005
0.053
0.002
0.000
0.000
0.001
1000
DiGress
0.028
0.046
0.008
0.027
0.032
0.047
0.009
0.025
0.037
0.046
0.069
0.051
500
SwinGNN
0.017
0.060
0.003
0.027
0.006
0.125
0.018
0,050
0.000
0.000
0.000
0.000
300
GGFlow
0.005
0.033
0.004
0.014
0.011
0.030
0.002
0.014
0.030
0.000
0.016
0.015
500
5.2 Molecule Graph Generation
We evaluated GGFlow on two standard molecular datasets, QM9 (Ramakrishnan et al. 2014 and ZIN2250k
[Iwwin et al. 2012), using several metries: Validity Validity without correction, Neighborhood Subgraph Pairwise Distance Kernel (NSPDK) Maximum Mean Diserepancy (MMD), and Frechet ChemNet Distance (FCD). To calculate these metrics, we sampled 10,000 molecules. We compared GGFlow against various molccule gencration models, including GraphAF, GraphDF, MolFlow (Zang & Wang 2020 EDP-GNN, GraphEBM (Liu et al. 2021), GDSS, PS-VAE (Kong et al., 2022), MolHF (Zhu et al. 2023). GruM, SwinGNN, DiGress, and GSDM. Detailed descriptions of the datasets, baselines and metrics provided in Appendix H
The results, presented in Table 2 indicate that GGFlow effectively captures the distribution of molccular data, showing significant improvements over the baselines. The high Validity without correction suggests that GGFlow successfully learns chemical valency rules. Additionally, GGFlow achieves superior NSPDK and FCD scores on both datasots, demonstrating its ability to generate molocules with distributions closely resembling those of natural molecules. Visualizations of molecules generated by different models are shown in Figure 2 with additional results on GGFlow provided in Appendix J
10

---

Published in Transactions on Machine Learning Research (12/2025)
Table 2: Generation results on the QM9 and ZINC250k datasets, Results are the means of 3 different runs. The best results and the second-best results are marked bold and bold.
QM9
ZINC250K
Method
Step
Val.
Val. w/o corr.
NSPDK
FCD
Val.
Val. w/o corr.
NSPDK
FCD
Training Set
100
100
0.0001
0.040
100
100
0.0001
0.062
GraphAF
100
67.14
0.0218
5.246
100
67.92
0.0432
16.128
GraphDF
100
83.14
0.0647
10.451
89.72
0.1737
33.899
MolFlow
100
92.03
0.0169
4.536
100
63.76
0.0468
20.875
GraphFRM
100
8.78
0.0287
6.402
100
5.29
0.2089
35.467
PS-VAE
0.0077
1.259
0.0112
6.320
MolHF
100
93.62
0.0387
23.940
EDP-GNN
100
47.69
0.0052
2.683
100
83.16
0.0483
16.819
1000
GDSS
100
96.17
0.0033
2.565
100
97.12
0.0192
14.032
1000
GSDM
100
99.90
0.0034
2.614
100
92.57
0.0168
12.435
1000
GruM
100
99.69
0.0002
0.108
100
98.32
0.0023
2.235
1000
SwinGNN
100
99.66
0.0003
0.118
100
86.16
0.0047
4.398
500
DiGress
100
98.29
0.0003
0.095
100
94.98
0.0021
3.482
500
GGFlow
100
99.91
0.0002
0.148
100
99.63
0.0010
1.455
500
Oo Okob
S4 whnts
ZINC250K
QM9
GFlow
DiGress
GruM
GDSS
Figure 2: Visualization of generated samples of different models in different molecular datasets
5.3 Conditional Generation
To further evaluate the performance of our model, we conducted conditional generation experiments on the QM9 dataset, focusing on generating molecules with molecular properties μ that closely match target value j². In the experiment, set the target value as 1, i.c 1
For the experiment, we employed reinforcement learning-based guidance method and compared it to the guided version of DiGress, which also proposes an effective approach for discrete diffusion models in conditional generation tasks. The reward function was defined as |/2 14=4"), and the model was trained over 10,000 steps using the training settings detailed in Section 5.2 To evaluate the effectiveness of our guidance method, we compared against three baselines: (1) Guidance for DiGress (Vignac et al. 2022). (2) Direct supervised training (ST) (3) Supervised fine-tuning (SFT). Additionally, we calculated the mean and variance of Іи 2 ffor samples generated unconditionally by both DiGress and GGFlow to provide baseline comparison. Further details of the experiment are provided in Appendix H..5
The results, detailed in Table 3 demonstrate the superiority of our reinforcement learning-based conditional generation method over both ST and SFT approaches. Notably, our method surpasses the guidance techniques used in diffusion models, showcasing its enhanced ability steer the generative process toward desired outcomes. Additionally, our approach achieves higher validity in conditional generated tasks, highlighting its robustness and superior performance in goal-directed generation.
11

---

Published in Transactions on Machine Learning Research (12/2025)
Table 3: Mean absolute error of molecular property μ on conditional generation on the QM9 dataset.
DiGress
GGFlow
Methods
Uncondition
Guidance
Unconditition
Supervised Training
+SFT
+RL
Mean
1.562
1.092
1.569
1.184
1.223
0.672
Variance
1.641
0.894
1.987
1.579
1.893
0.647
Val. w/o corr.
96.54
74.2
98.93
86.1
87.0
92.2
5.4 Ablation studies
To thoroughly assess the individual contributions of the proposed components to GGFlow's performance, conducted a comprehensive series of ablation experiments. These studies utilized both the Community-small and ZINC250k datasets. Our investigation focused on isolating the impact of: (1) the full GGFlow model, (2) the model performance without the integration of Optimal Transport (OT), denoted as GGFlow (w/o OT), (3) the model performance without the GraphEvo module, denoted as GGFlow (w/o Evo), and (4) the baseline GGFlow framework without either the GraphEvo module or Optimal Transport, designated as GGFlow (w/o both). The specific configurations for each ablated model are summarized in Table 4 The empirical results of these studies presented in detail in Table 5 Further specifics regarding the experimental setup are provided in Appendix 1.2
Table 4: Ablated model configurations used in the study,
Method
Flow Matching Framework
Optimal Transport
GraphEvo Module
DiGress
x
x
X
GGFlow (w/o both)
X
GGFlow (w/o OT)
X
GGFlow (w/o Evo)
X
GGFlow
As demonstrated in Table 5 a comparative analysis of GGFlow (w/o OT) and GGFlow reveals that the integration of Optimal Transport consistently improves generation performance. Notably, even in the absence of both GraphEvo and Optimal Transport, GGFlow (w/o both) consistently outperforms DiGress, thereby underscoring the inherent advantages of our flow matching framework over traditional diffusion models in graph generation lasks.
Furthermore, the GraphEvo module, particularly its triangle attention mechanism for edges, effectively captures more intricate node and edge features, leading to substantial performance enhancements. To investigate GraphEvo's ability to extract edge information, we calculated the cross-entropy of edges on the validation set for each epoch in Figure S4 We observed that after incorporating GraphLvvo, the edge cross- entropy decreased more rapidly, indicating the model's enhanced capability in capturing edge information.
To evaluate the benefits of Optimal Transport on sampling efficiency, we conductod experiments with varying inference steps, presenled in Figure 3 (More detailed comparison on Communiy--mall datasets in Figure S3). Our findings indicate that GGFlow achieves superior performance compared to other diffusion- based models with significantly fewer inference steps, particularly at 100 and 200 steps. This highlights GGFlow's enhanced sampling efficiency. Furthermore, the integration of Optimal Transport demonstrably boosts sampling efficiency and enhances generative performance across both generic and molecular graph generation tasks, as evidenced by the performance gains observed.
To demonstrate the enhanced training stability conferred by Optimal Transport and the Flow Matching framework, performed detailed analysis of evaluation metrics during the training phase. We compared
12

---

Published in Transactions on Machine Learning Research (12/2025)
Table Ablation studies on the OT and GraphEvo on the Commuinty-smal and ZINC250k datasets. Results are the means of 3 different runs. The best results are marked bold.
Sommunity-small
ZINC250k
Method
Step
Deg.
Clus.
Orbit
Val. w/o corr.
NSPDK
FCD
DiGress
0.032
0.047
0.009
94.98
0.0021
3.482
500
GGFlow (w/o both)
0.028
0.076
0.007
99.07
0.0022
2.104
500
GGFlow (w/o OT)
0.018
0.027
0.004
99.58
0.0015
1.503
500
GGFlow (w/o Evo)
0.015
0.055
0.005
99.12
0.0025
1.891
500
GGFlow
0.011
0.030
0.002
99.63
0.0010
1.455
500
Didress
GDSS
25
GDSS
GDFlow (wro
0.50
0.25
Average
15-
10-
0.20
0.35
255
1800
200
3000
Inforence
Inforance
(a) Community-salll
(b) ZINC250k
Figure 3: Ablation studies the impact of varying inference steps across Community-small and ZINC250k datasets. Performance of the ZINC250k dataset is evaluated using 1,000 generated molecules due to the computational cost of generating larger sets for metric calculation.
the average values on the Community-smal dataset and FCD values the ZINC250K dataset against DiGress, which shares similar training objectives. For a fair comparison and to highlight the distinct contributions, we specifically focused on GGFlow (w/o both) and GGFlow (w/o Evo). Test set average values and FCD were computed as described in Section 5.1 utilizing chockpoints saved every 100 training steps.
To quantitatively assess training stability, we calculated the Mean Absolute Difference (MAD) of evaluation metric values between adjacent epochs and averaged these differences. As depicted in Figure 4 the presence of Optimal Transport (comparing GGFlow (w/o Evo) to GGFlow (w/o both)) improved model stability, leading to more consistent performance during training.
Figure further illustrates that both GGFlow (w/o both) and GGFlow (w/o Evo) achieve faster and more effective convergence than DiGress. Importantly GGFlow (w/o Evo) consistently outperforms GGFlow (w/o both) in terms of both average metrics and FCD, clearly demonstrating the benefits derived from Optimal Transport. These findings collectively indicate that both the Optimal Transport module and the underlying Flow Matching framework significantly contribute to greater training stability.
6 Conclusion
In this paper, introduced GGFlow, discrete flow matching generative model for graphs that incorporates optimal transport and an innovative graph transformer network. GGFlow achieves statc-of-the-art perfor- mance in unconditional graph generation tasks. Additionally, we presented a novel guidance method using reinforcement learning to control the generative trajectory toward a preferred distribution. Furthermore, our model demonstrates the ability to achieve the best performance across various tasks compared to other basclines, which highlights the practical impact our guidance method. Despite these advancements, GGFlow has primary limitation in its scalability to large graphs (with more than 500 nodes). This is primarily
13

---

Published in Transactions on Machine Learning Research (12/2025)
DiGress (MAD: 0.037)
GGFlow wi both (MAD: 0.404)
6.5
Goflow both (MAD: 0.029)
GGPlow weo Eve (MAD: 0.413)
GGFfow Evo (MAD: 0.021)
Bvenge
PCD
61--
2003
5000
502233
Training
Community-small
(b) ZINC250K
Figure 4: Analysis of training stability for the Flow Matching framework and Optimal Transport on the Community-smi small and ZINC250k datasets. Test metrics of ZINC250k are calculated using 1,000 generated molecules.
due the computational complexity of the triangle altention updates and spectral feature compulations. Generation times for different graph scales provided in Appendix L.3 Another area for future work is to improve the thcoretical guarantees of our optimal transport method, as its current formulation only provides invariance to identical permutations rather than full permutation equivariance. Future research will focus on enhancing the scalability of GGFlow to larger graphs. We will also explore more generalized and efficient optimal transport methods that are fully foundation and broaden its applicability.
to strengthen the model's theoretical
7 Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants No. 82130055, 32271297, and 32370657; the National Key Research and Development Program of China under Grant No. 2020YFA0907000; the Key Special Project for Technological Innovation and Application Development of Chongqing Municipality under Grant No. CSTI2022TIAD-DEX0BS the King Abdullah University of Science and Technology (KAUST) Office of Research Administration (ORA) under Awards No. REI/1/5280 01-01, 101/115992-01-01, URF/ URF/1/6713 1671111- and FCC/1/5932-124 12-11- Center of Excellence for Smart Health (KCSH) under award number 5932; and Center Excellence on Generative AI under award number 5940. The data underlying this article are available from the first and corresponding author upon reasonable request.
14

---

Published in Transactions on Machine Learning Research (12/2025)
References
Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. Structured Denoising Diffusion Models in Discrete State-Spaces. Advances in Neural Information Processing Systems, 34:17981-17993, 2021.
Andreas Bergmeister, Karolis Martinkus, Nathanaël Perraudin, and Roger Wattenhofer. Efficient and Scalable Graph Generation through Iterative Local Expansion. arXiv preprinl (rXxi:2312.11222 2023.
Andreas Bergmeister, Karolis Martinkus, Nathanaël Perraudin, and Roger Wattenhofer. Efficient and scalable graph generation through iterative local expansion. In The Twelfth International Conference on Learning Representations, 2024.
Camille Bilodeau, Wengong Jin, Tommi Jaakkola, Regina Barzilay, and Klavs F Jensen. Generative models for molecular discovery: Recent advances and challenges. Wiley Interdisciplinary Reviews; Computational Molecular Science, :2(5):e1608, 2022.
Abraham Bookstein, Vladimir A Kulyukin, and Timo Raita. Generalized Hamming Distance. Information Retrieval, 5:353 375, 2002.
Joey Bose, Tara Akhound-Sadegth Kilian FATRAS, Guillaume Huguet, Jarrid Rector-Brooks, Cheng- Liu, Andrei Cristian Nica, Maksym Korablyov, Michael M Bronstein, and Alexander Tong. SE (3)-Stochastic Flow Matching for Protein Backbone Generation. In The Twelfth International Conference on Learning Representations, 2023.
Andrew Campbell, Joe Benton, Valentin De Bortoli, Thomas Rainforth, George Deligiannidis, and Arnand Doucet. Continuous Time Framework for Discrete Denoising Models. Advances in Neural Information Processing Systems, 95:28266-2822 2022.
Andrew Campbell, Jason Yim, Regina Barzilay, Tom Rainforth, and Tommi Jaakkola. Generative Flows on Discrete State- Spate-Spcces: Enabling Multimodal Flows with Applications to Protcin Co-Design. arXiv preprint 200.244900499 2024.
Liqun Chen, Ke Bai, Chenyang Tao, Yizhe Zhang, Guoyin Wang, Wenlin Wang, Ricardo Henao, and Lawrence reinforcement learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 7512 7520, 2020a.
Carin. Sequence generation with optimal-tannspot-enhaaan
Liqun Chen, Zhe Gan, Yu Cheng, Linjie Li, Lawrence Carin, and Jingjing Liu. Graph optimal transport for cross-domain alignment. In International Conference on Machine Learning, pp. 1542-1553. PMLR, 2020b.
Xiaohui Chen, Jiaxing He, Xu Han, and Li-Ping Liu. Efficient and Degree-Guided Guided Graph Generation via Discrete Diffusion Modoling. In Proceedings of the 40th International Conference on Machine Learning, pp. 4585 1685-4610, 2023.
2018 workshop on Theoretical Foundations and Applicutions of Deep Generalive Models, 2018. Nicola De Cao and Thomas Kipf. MolGAN: An implicit generative model for small molecular graphs. 1CML
Floor Eijkelboom, Grigory Bartosh, Christian Andersson Nacssoth, Max Welling, and Jan- Mal-Willem van de Meent. Variational flow matching for graph generation. arXiv preprinl arXiv:2406. 04.0000..44,.... 2024.
Paul Erdös, Alfréd Rényi, et al. On the evolution of random graphs. Publ. math. inst. hung. acad. sci, 5(1): 17-60, 1960.
Kilian Fatras, Younes Zine, Szymon Majewski, Rémi Flamary, Rémi Gribonval, and Nicolas Courty. Minibatch optimal transport distances; analysis and applications. arxiv preprinl orXii:2201.01 017, 2021.
Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky TQ Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching. ar preprint 2XXx:2207.11595, 2024.
Aditya Grover, Aaron Zweig, and Stefano Ermon. Graphite: Iterative Generative Modeling of Graphs. In International conference on machine learning, pp. 2434-244h PMLR, 2019.
15

---

Published in Transactions on Machine Learning Research (12/2025)
Kilian Konstantin Hacfeli, Karolis Martinkus, Nathanaël Perraudin, and Roger Wattenhofer. Diffusion models for graphs benefit. from discrete state spaces. In NewrIPS 2022 Workshop: New Prontiers in Gruph Learning, 2022.
Jonathan Ho, Ajay Jain, and Picter Abbeel. Denoising Diffusion Probabilistic Modcls. Advances in neural information processing systems, 33:6840 6851, 2020.
Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video Diffusion Models. Advances in Neural Information Processing Systems, 35:8633-86644 2022.
Xiaoyang Hou, Tian Zhu, Milong Ren, Bo Duan, Chumming Zhang, Dongbo Bu, and Shiwei Sun. Gtam: A molecular pretraining model with geometric triangle awarcness. Bioinformatics, pp. btac524, 2024.
Han Huang, Leilei Sun, Bowen Du, and Weifeng Lv. Conditional diffusion based on discrete graph structures for molecular graph generation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 4302 4311, 2023.
Md Shamim Hussain, Mohammed J Zaki, and Dharmashankar Subramanian. Triplet interaction improves graph transformers: Accurate molecular graph learning with triplet graph transformers. In Forty-fors Inter- national Conference Machine Learning, 2024. URL https:///penrrevion wttp://wwweeeeooooeeeemee http://wwww.......oo.ooooaaaaoa.
John B Ingraham, Max Baranov, 7ak Costello, Karl W Barber, Wujie Wang, Ahmed Ismail, Vincent Frappier, Dana M Lord, Christopher Ng-Thow-Hingg Erik R. Van Vlack, et al. Illuminating protein space with a programmable generative model. Nature, pp. 1-9, 2023.
John Irwin, Teague Sterling, Michael M Mysinger, Erin Bolstad, and Ryan Coleman. ZINC: A Free Tool Discover Chemistry for Biology. Journal of chemical information and modeling, 52(77)17577-176 2012.
Yunhui Jang, Seul Lee, and Sungsoo Ahn. A Simple and Scalable Representation for Graph Generation. In The Twelfth Internalional Conference on Learning Represendations, 2023.
Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction Tree Variational Autoencoder for Molecular Graph Generation. International conference on machine learning, pp. 2323-2223333 PMLR, 2018.
Jaehyeong Jo, Seul Lee, and Sung Ju Hwang. basee Generative Modeling of Graphs via the System of Stochastic Differential Equations. arXiv:2202.02511 02514, 2022a. URL https://arxiv.. org/ abs/////b/222.../////////2/ 02514
Jaehyeong Jo, Seul Lee, and Sung Ju Hwang. Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations. In International Conference on Machine Learning, pp. 10362-10383. PMLR, 2022b.
Jaehyeong Jo, Dongki Kim, and Sung Ju Hwang. Graph generation with diffusion mixture. In Forty-fort International Conference on Machine Learning, 2024.
John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneberger, Kathryn Tumyasuvuakkoo,, Russ Bates, Augustin Zidek, Anna Potapenko, et al. Highly accurate protein structure prodiction with AlphaFold. Nature, 596(7873):583--88 589, 2021.
Leon Klein, Andreas Krämer, and Frank Noé. Equivariant flow matching. Advonces in Neural Information Processing Systems, 36, 2024.
Lingkai Kong, Jiaming Cui, Haotian Sun, Yuchen Zhuang, Aditya Prakash, and Chao Zhang. Autoregressive diffusion model for graph generation. In International conference on machine learning, pp. 17391-17408. PMLR, 2023.
Xiangzhe Kong, Wenbing Huang, Zhixing Tan, and Yang Liu. Molecule generation by principal subgraph subgraap mining and assembling. Advances in Neural Information Processing Systems, 35:2550 2563, 2022.
16

---

Published in Transactions on Machine Learning Research (12/2025)
Hayeon Lee, Eunyoung Hyung, and Sung Ju Hwang. Rapid Neural Architecture Search by Learning to Generate Graphs from Datasets. In International Conference on Learning Represenlations, 2020.
Yujia Li, Oriol Vinyals, Chris Dyer, Razvan Pascanu, and Peter Battaglia. Learning Deep Generative Models of Graphs, arXiv preprint ar arX3: 03324, 2018.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hame Maximilian Nickel, and Matthew Le. Flow Matching for Generative Modeling. In The Eleventh International Conference on Learning Representations, 2022.
Jenny Liu, Aviral Kumar, Jimmy Ba, Jamie Kiros, and Kevin Swersky. Graph Normalizing Flows. Advances
Neural Information Processing Systems, 32, 2019.
Meng Liu, Keqiang Yan, Bora Oztekin, and Shuiwang Ji. GraphEBM: Molecular Graph Generation with Energy-Bess Models. arXiv preprint arXiv:2102.005: 00546, 2021.
Shiwei Liu, Tian Zhu, Milong Ren, Chungong Yu, Dongbo Bu, and Haicang Zhang. Predicting mutational effects on protein-protein binding via a side-chain diffusion probabilistic model. Advances in Newral Information Processing Systems, 36, 2024a.
Yijing Liu, Chao Du, Tianyu Pang, Chongxuan Li, Wei Chen, and Min Lin. Graph Diffusion Policy Optimization. arXiv preprinl or Xix:22202 16302, 2024b.
Tianze Luo, Zhanfeng Mo, and Sinno Jialin Pan. Fast Graph Generation via Spectral Diffusion. IEEE Trunsactions Pallern Analysis and Machine Intelligence, 2023.
Youzhi Luo, Keqiang Yan, and Shuiwang Ji. GraphDF: A Discrete Flow Model for Molecular Graph PMLR, 2021.
Generation. In International conference on machine learning, pp.
Tengfei Ma, Jie Chen, and Cao Xiao. Constrained Generation of Semantically Valid Graphs via Regularizing Variational Autoencoders, Advances in Neural Information Processing Systems, 31, 2018.
Manuel Madeira, Clement Vignac, Dorina Thanou, and Pascal Frossard. Generative modelling of structurally constrained graphs. arXio preprint or arXiv:24066 (6. 1941, 2024.
Giorgia Minello, Alessandro Bieciato, Luca Rossi, Andrea Torsello, and Luca Cosmo. Graph generation via spectral diffusion. arXie preprint orXim:2402.1897, 2024.
Chenhao NIu, Yang Song, Jiaming Song, Shengjia Zhao, Aditya Grover, and Stefano Ermon. Permutation Invariant Graph Generation via Score- Based Generative Modeling, In International Conference on Artificial Intelligence and Statistics, pp. 4474-4484. PMLR, 2020.
Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almcida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. Training language models to follow instructions with human feedback. Advances in newral information processing systems, 35:27730-277444 2022.
Hermina Petric Maretic, Mireille El Gheche, Giovanni Chierchia, and Pascal Frossard. Got: an optimal transport framework for graph comparison. Advances in Neural Information Processing Systems, 32, 2019.
Alexanle Pooladian, Heli Ben-Hamu, Carles Domingo-Enridh Enrich, Brandon Amos, Yaron Lipman, and Ricky TQ Chen. Multisample flow matching: Straightening flows with minibatch couplings. arXiv preprint 1477772304.4477, 2023.
Yiming Qin, Manuel Madeira, Dorina Thanou, and Pascal Frossard. Defog: Discrete flow matching for graph generation. arXiv preprint 1rXi::2400. 04263, 2024.
Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molccules. Scientific data, 1(1):1-7, 2014.
Milong Ren, Tian Zhu, and Haicang Zhang. Carbonnovo: Joint design of protein structure and sequence using unified energy-based model. In Forty-first International Conference on Machine Learning, 2024. URL htt.htttss///opeeeeee.cc http://www........o...m///////aa
17

---

Published in Transactions on Machine Learning Research (12/2025)
Ida Schomburg, Antje Chang, Christian Ebeling, Marion Gremse, Christian Heldt, Gregor Huhn, and Dietmar Schomburg. Brenda, the enzyme database: updates and major new developments. Nucleic acids research, 32(suppL_1):D433) -)33433--D33,, 2004.
Prithviraj Sen, Galileo Namata, Mustafa Bilgie, Lise Getoor, Brian Galligher, and Tina Eliassi-Rart Collective Classification in Network Data. AI magazine, 29(3):93-99 93, 2008
Chence Shi, Minkai Xu, Zhaocheng Zhu, Weinan Zhang, Ming Zhang, and Jian Tang. GraphAF: Flow- based Autoregressive Model for Molecular Graph Generation. In International Conference on Learning Representations, 2019.
Martin Simonovsky and Nikos Komodakis. GraphVAE: Towards Generation of Small Graphs Using Variational Autocncoders. In Artificial Neural Networks and Machine Learning-ICANN 2018: 27th International Conference on Arlificial Neural Networks, Rhodes, Greece, Oclober 4-7, 2018, Proceedings, Part T 27, pp. 412-422. Springer, 2018.
Daniel GA Smith, Lori A Burns, Andrew C Simmonett, Robert M Parrish, Matthew C Schieber, Raimondas Galvelis, Peter Kraus, Holger Kruse, Roberto Di Remigio, Asem Alenaizan, et al. PS14 1.4: Open-source software for tigu-throughput quantum chemistry. The Journal of chemical physics, 152(18), 2020.
Yang Song, Jascha Johl-Dickstein, Diederik Kingma, Abhishek Kunar, Stefano Ermon, and Ben Poole. Score-Based Generative Modeling through Stochastic Differential Equations. In International Conference Learning Representations, 2020.
Yuxuan Song, Jingjing Gong, Minkai Xu, Ziyao Cao, Yanyan Lan, Stefano Ermon, Hao Zhou, and Wei- Wing Ma. Equivariant Flow Matching with Hybrid Probability Transport for 3D Molecule Generation. Advances Neural Information Processing Systems, 36, 2024.
Alexander Tong, Jessie Huang, Guy Wolf, David Van Dijk, and Smita Krishnaswamy. Trajectorynet: A dynamic optimal transport network for modeling cellular dynamics. In International conference on machine learning, pp. 9526 10369553.666 PMLR, 2020.
Alexander Tong, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Kilian FATRAS, Guy Wolf, and Yoshua Bengio. Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport. In ICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems, 2023.
Titouan Vayer, Laetitia Chapel, Rémi Flamary, Romain Tavenard, and Nicolas Courly. Optimal transport for structured data with application graphs. arXiv preprint arXiv:1805.09114 2018.
Clement Vignac, Igor Krawczuk, Antoine Siraudin, Bohan Wang, Volkan Cevher, and Pascal Frossard. Digress: Diserete denoising diffusion for graph generation. arXiv preprint 2XNi:2209.1473, 2022.
Cédric Villani. Optimal Transport, volume 338 of. Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences], pp. 71, 2009.
Bram Wallace, Meihua Dang, Rafael Rafailov, Linqi Zhou, Aaron Lou, Senthil Purushwalkam, Stefano Ermon, Caiming Xiong, Shafiq Joty, and Nikhil Naik. Diffusion model alignment using direct preference optimization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8228 8238, 2024.
Hongwei Wang, Jia Wang, Jialin Wang, Miao Zhao, Weinan Zhang, Fuzheng Zhang, Xing Xie, and Minyi Guo. GraphGAN: Graph Representation Learning With Generative Adversarial Nets. In Proceedings of the AAAI conference on artificial intelligence, 2018
Joseph L Watson, David Juergens, Nathaniel Bennett, Brian L Trippe, Jason Yim, Helen E Eisenach, Woody Ahern, Andrew J Borst, Robert Ragotte, Lukas F Milles, et al. De novo design of protein structure and function with RFdiffusson. Nature, 620(7976):1089-1100 2023.
18

---

Published in Transactions on Machine Learning Research (12/2025)
Zhe Xu, Ruizhong Qiu, Yuzhong Chen, Huiyuan Chen, Xiran Fan, Menghai Pan, Zhichen Zeng, Mahashweta Das, and Hanghang Tong. Discrete-state continuous-time diffusion for graph generation. or Xie preprinl ar arXiu:405.11441, 2024.
Qi Yan, Zhengyang Liang, Yang Song, Renjie Liao, and Lele Wang. Swingnn: Rethinking permutation invariance in diffusion models for graph generation. arXiv preprint X01:22307.01646, 2023.
Jason Yim, Andrew Campbell, Andrew YK Foong, Michael Gastegger, José Liménez-Luna Sarah Lewis, Victor Garcia Satorras, Bastiaan S Voeling, Regina Barzilay, Tommi Jaakkola, al. Fast protein backbone generation with SE (3) flow matching. ar preprinl arXiv:2310. 000:231000597, 2023.
Jiaxuan You, Rex Ying, Xiang Ren, William Hamilton, and Jure Leskovec. GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models. In International conference on machine learning, pp. 5708-5717 PMLR, 2018.
Chengxi Zang and Fei Wang. MoFlow: an invertible flow model for generating molecular graphs. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Рp. 617-626, 2020.
Lingxiao Zhao, Xueying Ding, and Leman Akoglu. Pard: Permulatioon-invaiaal auloregressive diffusiin for graph generation. arXic preprint (.ki:2202200687,, 2024.
Tian Zhu, Milong Ren, and Haicang Zhang. Antibody design using a score-based diffusion model guided by evolutionary, physical and geometric constraints. In Forty-first International Conference Machine Learning, 2024. URL htttp::/ wttp://wwwwe//footrteeeoeee nett/forumiid-1ys000045
Yiheng Zhu, Zhenqiu Ouyang, Ben Liao, Jialu Wu, Yixuan Wu, Chang- Yu Hsieh, Tingjun Hou, and Jian Wu. Molhf: a hicrarchical normalizing flow for molecular graph generation. In Proceedings of the "hertt-Secood International Joint Conference Artificial Intelligence, pp. 5002-5014 2023.
19

---

Published in Transactions on Machine Learning Research (12/2025)
Appendix
A Background
A.1 Continuous Flow Matching Generative Model
The gencrative model aims to establish a mapping f IR Rd that transforms a noise distribution go into targel data distribution qi- This transformation is dependent on density function Po over R2, and integration map 1h,, which induces pushforward transformation Pt [ (Po). This denotes the density of points x Po transported from time O to time t along vector field u [0, 1] x R - -R.
The vector field u is formulated as:
dx = (z(ddt
The solution (x) to this ODE, with the imitial condition vo(x) = x, represents the trajectory of the point æ governed by from time to time t.
The evolution of the density Pt, viewed as function p: |0, x Rd + R, is encapsulated by the continuity equation:
dp = (pene), Ōt
with the initial condition given by Po- Here, u is the probability flow ODE for the path of marginal probabilities P. generated over time.
In practical applications, if the probability path pe(x)) and the generating vector field We (x) are known and (x) tractably sampled, we leverage a time-dependent neural network vo(.-) [0, 1] x Rd Rd to
approximate u. The neural network trained using the flow matching objective:
(F) Be..ue..)..epcccelllll
(x)))
(19)
which enhances the model's capability to simulate the target dynamics accurately. Avoiding the explicit construction of the intractable vector field, recent works express the probability path marginal over
joint involving latent variable p(xz) (x=l)) (Lipman et al. 2022 Tong et al. 2023) and the (=)=(2))2) is conditional probability path, satisfying some boundary conditions at t t= 0 and
The conditional probability path also satisfies the transport equation with the conditional vector field u (x(x1)
(zlzz))
(20)
Ø1
We can construct the marginal voctor field us (x) via the conditional probability path Pep (2r(21) as:
u (z) = Egy [u (x(21))
(21)
We can replace the flow matching loss LPM with an equivalent loss regressing the conditional vector field (x(1)1)) and marginalizing 2i instead:
(CF = Bentsa.),
us (x(x1))
(eLFM(O) VeCCFM(O).
So we can use (f) instcad train the parametric vector field ug-
B Related Works
B.1
Comparison with Discrete Flow Matching
Campbell al. (2024) first introduced flow matching in discrete spaces using continuous-time Markov chain.
Building on this. Gat al. (2024) expanded the framework to encompass general source and target couplings,
20

---

Published in Transactions on Machine Learning Research (12/2025)
inchuding U-coupling and C-coupling. GGFlow advances discrete flow matching and its source-target coupling to suit graph-structured data. Our approach innovatively incorporates efficient optimal transport for graphs within the flow matching framework. To address the inherent sparsity and permutation invariance of graphs we employ a tailored prior distribution. Additionally, we implement an edgo-augmented graph transformer enhance generative performance and adopt goal-guided framework for conditional generation. These advancements collectively enhance the practical applicability of GGFlow.
B.2 Comparison with Graph Discrete Diffusion Model
DiGress (Vignac el al. 2022 and PPGN (Haefeli et al. 2022) were among the first to apply discrete diffusion models to graph generation, highlighting the advantages of discrete state spaces. DiGress further introduced
optimal prior distribution and global structural features specifically designed to enhance graph generation. Their forward and generative processes are expressed as:
q( Cat , == Q)Q:), with Q = Q1Q2...@@ QQ.
(22)
Cat(GG--ipp G'QT o - G
GG
(23)
where G represents the noisy graph at time i, and Qe is the time-dependent transition matrix. These methods require maintaining convergence properties of the transition mairix and cumulative matrix products, constraining the choice of prior distributions and destabilizing training. In contrast, GGFlow employs a simpler interpolation between the prior and data distributions during training, avoiding cumulative products and improving both training stability and the ease of selecting appropriate priors.
B.3 Comparison with Graph Discrete Flow Model
GraphDF (Luo et al. 2021 uses discrete flow model to generate molecular graphs by sequentially sampling discrete latent variables and mapping them to nodes and edges via invertible modulo-shift transforms. GGFlow simplifies this by transforming the invertible modulo-shift into a conditional vector field that interpolates between the prior and data distributions, bypassing the need for complex invertible mappings. Furthermore, while GraphDF adopts an autoregressive process for graph generation, GGFlow generates the entire graph in one-shot manner, capturing holistic relationships among nodes and edges more efficiently.
B.4 Comparison with Graph Flow Matching Models
CatFlow [Bijkelboom al. 2024 employs variational inference to apply flow matching to categorical data, but it only considers the conditional vector field under the assumption of independent coupling in the joint distribution x (GG, G') and fails to consider the inherent sparsity of graph structures. and DeFoG (Qin et al. 2024) also applies the discrete flow matching to establish the generation process of the graph, while they don't consider the generalization the joint distribution. GGFlow extends this by generalizing (Go G!) as an efficient Wasserstein optimal transport map and incorporating an optimal prior distribution tailored for graph structures, improving performance in generation tasks. Additionally, GGFlow introduces a novel goal-directed approach for discrete flow matching in conditional generation tasks, enhancing its practical applicability.
B.5 Relationship to DiGress
DiGress and GGFlow are both generative models that share similar training objectives, specifically denoiser prediction. In contrast, their mathematical derivations and conceptual starting points differ.
DiGress, a diffusion model, is trained to predict the original data point (G") from a noisy version (G*). Its training objective, DDIGress: derived from the evidence lower bound (ELBO) of the data likelihood, which is gradually loosened a cermutation-ivariiant loss, as detailed in Lemma 3.2 of Vignac al. (002)). The objective is defined
21

---

Published in Transactions on Machine Learning Research (12/2025)
DiGGres
- P (oo(logo(GG(G)
Epaaaa (G0) log S CDT
E Pannn C
k=1
(DT EB peata (Go)
KL (L(4 (pror( 1o [Logg oll (G"IG))
K-1
,, "a ( (CC
E
The full derivation of the training objective and inequality, detailed Proposition in Campbell et al. 2022 shows how the ELBO manipulated to arrive at. this loss function.
GGFlow, a flow-matching model, also uses a denoiser prediction objective, but its derivation is rooted in different principles. The training objective, LOGFLow: is derived from an ELBO of the flow now-matching process, detailed in Section C.2 of Campbell al. (2024):
LGGFLOW =Rypaaca (tw.:///w..........o.co.o..oooo. E paaua (Gt) (- log Po loggo
G9)]
(G S - LELBO = - -LGGFlow + Ln + fKL + C) S - (LooFlow + Ln + C).
(G LR - (a (Rr log oo [G'
Unlike DiGress, the GGFlow ELBO includes an additional term, LR. As argued in Section C.2 of Campbell et al. (2024), this term is often deemed unnecessary as it encourages the generative rate to match a chosen jump rate, simplifying the final training objective to just. LGGFLOW- Fortunately, this simplified training objective also satisfics permutation invariance, a shared desirable property with DiGress.
C Proofs
C.1
Optimal Prior Distribution
This prior is structured as a product of a single distribution v for nodes and a single distribution e for all edges, II, v II e, to ensure exchangeability across the graph components.
Theorem (Optimal prior distribution). Consider the class C (II, u x II.s v, (u, E P(V) x P(E)} of distributions over graphs, which factorize the product of uniform distribution over node attribude space V and uniform distribution over edge attribute space E. Given any arbitrary distribution P over graphs (viewed as lensor of order n n²²,, with qv and qE as ils marginal distributions for node and edge allribules respectively, then the orthogonal projection of P onto C is defined as @c = II: gv x Ilis 9E. This projection minimizes the Euclidean distance:
arg min |P (0.e)EC
II UX
e
The details and proof of Theorem are extensively discussed DiGress (Vignac el al. 2022
C.2 Choice of conditional velocity field
The subsequent theoretical discussion is structured into three parts, each establishing a fundamental result. necessary for our proposed framework:
Lemma demonstrates that the vector field u, (G", G), defined as an expectation over the conditional vector field u,(G", G|Go, G') (Equation 11], satisfics the Kolmogorov forward equation, provided the conditional rate adheres to the conditional Kolmogorov equation. This finding establishes the fundamental consistency of the vector field and extends Proposition 3.1 in Campbell et al. 2024)).
22

---

Published in Transactions on Machine Learning Research (12/2025)
2. We then introduce specific functional form for the conditional vector field. Proposition 3 proves that this proposed formulation satisfies the conditional Kolmogorov equation. This result. is an extension of Proposition 3.2 found in Campbell et al. (2024).
3. Finally, for practical implementation and computational efficiency, Corollary ! presents simplified, implementable expression for the conditional vector field, derived directly from Proposition B
The specific points of departure between our proof techniques and the original proofs by Campbell et al. (2024) are indicated using blue text for ease of reference and comparison.
Lemma 1. If (G" G|G", G!) a 1Laevve ue(x", a x/ x') is a conditional vector field that satisfies the conditional Kolmogorov equation
O.P:n (z lot, x²)= =
Σ
ue(0", x0/" xt pen (al2 ,0)) - Σ
u1(x,, 2/22 x² pen (2'lat,
(24)
for each single generic random variable x E VUE, then the vector field
u(GG, G*) A (p G))) = II
Pe (ao) xla",, x,)))
(25)
XEVUE
also satisfies the Kolmogorov equation Ospr(r²)= =E Us p))))(()) Exfrr uela' p)pen()))
Proof. We assume the graph generation process can be modeled by a time-inhhomoeeeouus Markov Chain, where the component-wise transition is defined by the conditional vector field (z(",, x>>>, x²).. Our goal is derive the marginal vector field ux(x",)) .2) and confirm that the marginal distribution pe(x*) satisfics the Kolmogorov forward equation. We use the fact ue(x", ; 2x120" x=t) Extz, ur we(a", se/e ,-²) for compactness.
We denote the joint distribution the initial state G and the terminal state G as (G G*), which implics म (x=", xx²) for a generic variable x E VUE.
O.P: (x'1 ,,,,,,0))= ;=0)= = Σ
ue uela" x//=" (x)))) 229)
x
E 20) (0PPe] (a" |x- x00] =Ea(ut w0)
E ue (x' sl/= x² Pat (olx² ,x0)]
0, (1)1)) (2 |x =²)== = Σ Σ T(x x (ul', s// al )ap (x/x=) ,0)
Using the identity x(x", x )ep (x|x² ,,0) pr (x)) [r(r (derived from Bayes' Rule where Pije(e', xz(z))
is the posterior distribution of the endpoints given the intermediate state z):
0epr(x')= = Σ Σ
Pije(e 22 (a", "/a" o/10, )x((x) based on Bayes Rule
(x9,21)
0-p((x')= = Σ(
==")at=="=))u((( 2/)= (xx)
(
0ppi(x')= = Σ ue (z", (z)
The intermediate steps demonstrate the specific algebraic transformation required for modified coupling, which represents a key point of departure from Campbell et al. (2024). Since the overall graph transition is assumed to be component-wis independent, the vector field u. ( G) also satisfies the Kolmogorov forward equation.
Proposition 3. We then prove that the proposed conditional vector field,
ReLU(0ppp (x/2² x²)) Oepen (2" |z x,²))
L. G G+)= G
u,(x 20/201 x²)) SEVUE
II
II
(26)
Z, Pel1 (x x x0)
#EVUE
23

---

Published in Transactions on Machine Learning Research (12/2025)
satisfies the conditional Kolmogorou equation. where ReLU(() max(a, 0) and Z = K( pe(x"1=0,2"
,x') > 0}|:
u1("", ar/= x. = 0 when (r( x²) = 0 and pe(x'|a* x²)) O. When of = x, the conditional vector field ue(a", x² x²) = - -x ue z|o0 x²). We assume that if Ptp (2|/2) = 0, then depap (x|x),, =0
Proof. The proof proceeds by verilying that the conditional Kolmogorov equation is satisfied by the conditional vector field uz( x=(""x²") for each single generic random variable I E VUE.
This conditional vector field, inspired by discrete flow matching, is defined as:
" x/x0 x=)x",=')==)) = (1 x²) ( 2x))
Z. Pap ( (xlx',=0)
where Z, = Hx² (=(x'==0,,' 03| is normalization constant. We first consider the case where Peji (eji (z* "tt//=",_a0, (=²,,²) > O. the Kolmogorov Equation requires: LHS = dipiji (x,,=") The Right-Hand Side (RHS)
is defined by the flux balance:
(OLP( x²) Dept (2]2= x²))
RHS
Σ
Pejz (x0l 2z0)
Z, Z..Pel Me.Puli (xrla, x0))
axfar Pejt (or 33)00
ReLU(O:P:n (a) 1 x²)
lxt (x,)))))))))))
Σ
(af1(0011> 120 (22))=) 20)
Z, Pr| (a* [el| x0)
2z.z (1 20)0
Σ
ReLU(0(Pp) (x' xx))= - depep (x|ze, 20))
##2" Pijz (2)e
Σ
ReTU(O0pi(ala"" ,x))
Zi
z iPes (eelat w0)>0
Σ
(OrPen 200) ) (xpe) 20))
Zi
# 2 20)00
Z. 1
OePej1 (x0
Σ
Z4
nta Peti .=))=
Z. -
1
Oepuy1 (I') (=")=")00))
0, Peja (x |xx (xt,,²))
Z.
Z.
Z. dePtli (z01ax",00 200) )
0epr ( 20)
+ Z,
Z₁
-.pe. (xx 12 ,,00) 2=0)= -LHS
In the case that Pet1 (x" |x _ 0 by assumption we have that depen (1")2), = 0. We have both
u(xx,, x = = and ue (x, x² |z=0 x-) = 0 because Piji (#f|x , = O. Therefore we have LHS = RHS 0
and thus the Kolmogorov equation is satisfied
Corollary 1. We can simplify the conditional vector field in the Proposition 3 as:
ReLU(ddpr (x/x xx)) depen (x (x'|=), |
(1 G. G))=
II DEVUE
uc(x 22)=",1))
II JEVUE
Z. Pej (xf 1x x,))
8(xt,20) 20)
II ZEV
(1 (1+ -
7,(1 t)
where ReLU(a)) = max(a, 0) and Z. [x pe(x|=0,x') > 0.r E VUEJI. ux(x', x//0 when
Ptj1 (zd 1x-, x00) =0. When x² x, the conditional vector field ue(at 2/1 (=,,=t))))))))))))))) En' x ,
24

---

Published in Transactions on Machine Learning Research (12/2025)
Proof. ( the factorization Pr(G") II, IL p(ejx). This implies that the conditional probability path and the resulting vector field decouple over the graph components. We therefore derive the rate matrix for single generic random variable I E U (representing any specific node or edge).
Consider the conditional probability path Pt)(2)) = tō{2, x') + (1 t)6{,, x") We define the time-derivative vector field v(x) (x)))(x). Differentiating with respect to yields:
if x² i x. otherwise.
v(z) = = x(xx
-1
(27)
We define the numeralor of the rale equation as N(x x) = ReLU(((()) - v(x²)) We analyze N based on the value of the current state I², Since the probability path Pap Pap(2) (2) concentrates all probability mass on the interpolation between noise x and data x!, restrict analysis to these two supported cases:
Case I: x' = x'. Here v(x') = 1. Since v(x) S N NN N =0
Case II: x¹ 2º. Here v(x²) -1 The term is ReLU(u(2)) +
If moving to target (r = x²): =2
If staying (xr 2²): N 0.
If moving to noise (x + x², x²); N = 1.
This expressed as: N ,(x,,=)= 8(2,
Finally, we consider the denominator D(x')= Z, Pıjz (x' |xt" a²). The flow is non-trivial only in Case II, where Piji [a"]=",,"" 20) - t. Combining the numerator from Case II with this denominator yields the womponent-wise rate:
8(x',
ux(x', x/20 2l122x"1) xx) Z.(1-t) 1(=,0'x' -(x,,=1))
(28)
Since the noise structure factorizes over the graph components (nodes and edges) as shown Equation 7 the gencration process equivalent to solving independent flow matching problems for cach component. Consequently, the vector field for the entire graph G, denoted (G',, G|G G*), is not scalar field but collection of independent component wise rates.
We define the global rate tensor 4 (G', "G', G| ( clement-wise for all clements in the sets V and E:
)(z1,=²) Z, - t)
u((x,, x/20 "11",,=*=== , " G G A II
II SEVUE
(1 (1+ (1+ o(x,
(29)
zEVUE
C.3 Proof of Proposition
Proof. The Kolmogorov forward equations for discrete flow matching are expressed as:
O.p: = upt:
(30)
If we establish the permutation invariance of the prior distributions Pref and the permutation equivariance of conditional flow probabilities, then it follows that p(G²) is permutation exchangeabl..
According Theorem 2 we deduce the permutation invariance of the prior distribution Pref- Given the conditional probabilities p(G'+A((G) 5(G++A G9 + (G GL+At)AI, it suffices demonstrate the
25

---

Published in Transactions on Machine Learning Research (12/2025)
permutation equivariance of the conditional probabilities. This requires showing the permutation equivariance of the vector field Consider the case for nodes:
oul v (V% vetar) (M (V), (V,,
(V),
LHS uV' (V--(a) ())()))
RHS (EpY,.r Vo 10) Iuy Iuy (V² 1(1) Ve --1
VV 1(1) v2 mmll
(V2 (V²-10) (() LHS.
where is permutation operator. This establishes the permutation equivariance of and the exchangeability of the generated distribution.
C.4 Proof of Theorem
First, we want to clarify the rationale and foundation of our theorem. The goal of optimal transport is to pair source and target data points with minimal cost during training, which is beneficial for our interpolation [Bose et al. 2023 Song et al. 2024). Thus, we design our optimal transport approach from the perspective of interpolation.
We define the node order of the graph G as the order of the nodes and edges in matrix representation. For example, if the node set of G is B, the possible node orders include (A, B, C), (B, or (.
In the interpolation process, we transform the graph representation to a matrix representation before performing interpolation. For example, for source data G = (Vo, E²), v E [ E e gaxaxm and target V E Rax= E¹ Raxaxm, where
data G of nodes, and m is the class number of edges, the node orders of Go and G have been fixed. Therefore, interpolation is performed directly on these fixed node orders.
the mumber of nodes, is the class number
The optimal transport aims to find pairs with the minimum cost for interpolation, and the interpolation is conducted on a fixed node order. Additionally, during optimal transport calculations, we also utilize the matrix representation of these graphs and our prior distribution is permutation invariant. Therefore, we aim to match source data with the target data G whose node order fixed, to achieve minimal transport cost. Furthermore, we assume that all pairs of source and target data share the same node order during optimal transport, which also facilitates the identification of pairs with minimal cost.
Regarding the permutation of the intermediate graph Go, we have oG lo + (1-1)oG', where G and G share an identical permutation. Our network & (G" G*) needs to maintain permutation equivariance, such that p² (G Ga oG*) = aBt (G" G²) for any permutation a to approximate oG So we prove the invariance of optimal transport under identical permutations, ie. ( = T ,GG").
Proof. Building on the foundations established in Theorem 2 and Proposition 1 we confirm the permutation invariance both the target and source distributions. The Hamming distance invariance under identical permutations n, as shown by:
H(G ) Σ 8(0f,u))
1 Σ e(e))
b
600 10 100) + 1 Σ
8(62 >olee 2 i
"iy o 103 )
H(oG" o
This property of the Hamming distance ensures the invariance of the optimal transport map under identical permutations.
26

---

Published in Transactions on Machine Learning Research (12/2025)
Additionally, the prior distribution is permutation invariant and our GraphEvo is permutation equivariant, all permutations of graphs are generated with equal probability (Fijkelboom et al. 2024).
Lemma 2. Let GG) be exchangeable distribution and our model Puje(G Go) is permulation equimari- ant. Then, all permutations of the generated graphs are selected with equal probability.
Proof. As the permutation equivariance of our model Pnn(G G²), implics the equivariance our vector fields u, Moreover, the sampling procedure exhibits permutation equivariance, where и is a permutation.
lir(G"oo"', = ofnj G"); == Po oG")
oo )t oo*
= = GO) + uo ( G "" = PPANIOGAMI G)) t=0
PHAAI (G oG o öl0 oG*) + u (GAHt) oo o0"))
= G*] + (G (GIG+,,," ()) = OPIAA (
t= = .,, At
Therefore, since po assigns equal density to all permutations of G, the resulting distribution preserves this property.
D Details of GraphEvo
GraphEvo is a novel edge-augmented graph transformer model designed for graph data, To enhance the generative capabilities of GGFlow, GraphEvo introduces a triangle update mechanism, which significantly improves the exchange of edge information. We incorporate FiLM and PNA layers into our architecture
[Vignac et al. )022):
FiLM(X, X2) = X2 (Lincarr(Xx) (22) + Lincar((X2)
PNA(X) = Linear Cat(maa(XX) min(X), mean(m)))..... std(x)))
The time complexity of GraphEvo is O(N3).
GraphEvo integrates global structural features to improve generation performance, including both graph- theoretic and domain-specific attributes:
Those encompass node-level propertics such as the number of k-cycles (k S 5) containing this point and an estimate of the largest connected component, alongside graph-level metrics like the total number of k-cycles (k < 6) and connected components.
Graph-thcorctic features:
Molecular features: These account for the current valency of each atom and the molecular weight of the entire molecule.
D.1 Proof of Proposition 2
Proof. Let G (V', F+) is a intermediate graph, and aG (o*V,o"Fo is the permutation. To prove the permutation properties of the graph, we nced consider two aspocts: additional structural features and the model architecture.
First, the spectral and structural features permutation equivariant for node-level features and invariant for graph-level features. Additionally, the FILM blocks and Lincar layers are permutation equivariant, while the PNA pooling function is permutation invariant. Layer normalization is also permutation equivariant.
As GraphEvo is built using permutation equivariant components, we conclude that the overall model is permutation cquivariant.
27

---

Published in Transactions on Machine Learning Research (12/2025)
E
Analysis of Optimal Transport
To analyze the time complexity optimal transport (OT), we compared the training time of OT with that of DiGress, using identical architectures on an NVIDIA A100 80G GPU. We evaluated the effects of model size, batch size, and number of nodes by measuring the duration of single training steps across three different datasets. Our results indicate that the time required for OT accounts for only 5% of the total training time, highlighting the efficiency of our optimal transport.
Table S1: Time Complexity of Optimal Transport
Dataset
Planar
Zinc250k
Community-small
DiGress Training Time (s)
0.1647
0,1690
0.0456
GGFlow Training Time (s)
0.1264
0.1301
0.0408
Optimal Transport Time (s)
0.0025
0.0070
0.0024
Percentage of OT
1.9%
5.3%
5.6%
Model Size (M)
3.6
4.6
6.4
Batch Size
64
128
80
Number of Nodes
64
[6,38]
[12,20]
To further demonstrate the advantages of our proposed distance metric, we conducted comparative analysis using F fused-Gromose Wasserstein (FGW), which is defined in Vayer et al. (2018), as an alternative permutation-invarian graph distance for model training.
A primary challenge encountered with FGW was computational overhead. We observed that calculating the FGW distance on both the Community-small and ZINC250K datasets required several orders of magnitude more time than our model's typical training duration. As detailed in Table S2 the computation time for FGW ZINC250k was particularly prohibilive, preventing us from completing full training run for comparative model. Consoquently, a comprehensive model trained with FGW was only feasible on the Community-small dataset.
Table S2: Time comparison of Optimal Transport distance computation
Dataset
Zinc250k
Community-small
Hamming Distance Training Time (s)
0.0070
0.0024
Fused Cromov-wassersterr Distance Training Time (s)
60.3062
9.0301
Table S3
presents the generative performance comparison on the Community-sni small dataset. Our model, utilizing the Hamming distance, outperformed the FGW-based model on the Cluster and Orbit metrics, while exhibiting slightly reduced performance on the Degree metric.
Table S3: Generative results with different Optimal Transport distances on Community datasct
Method
Degrce
Cluster
Orbit
Hamming Distance
0.011
0.030
0.002
Fused wasseeewasserstein Distance
0.004
0.069
0.006
More critically, our model demonstrated superior training stability compared to the FGW-based model, as evidencod by the validation KL divergence on the validation set. Figure [ST illustrates this enhanced stability. This improved stability is directly attributable our distance metric being specifically tailored and better adapted to the Flow Matching training methodology.
28

---

Published in Transactions on Machine Learning Research (12/2025)
5.0
DiGress (MAD: 0.02)
GGFlow with Graeooorrooo (MAD: 0.04)
GGFlow with Hamming Distance (MAD: 0.01)
4.5
Validation KL divergence
4.0
3.5
3.0
2.5
2.0
0
1000
2000
3000
4000
5000
Steps
Figure S1: Validation KL Divergence comparison of different Optimal Transport approaches on Commmity- small dataset
These findings collectively indicate that, while our proposed optimal transport approach limits invariance to identical permutations, it does not negatively impact the model's practical performance and is better suited and more efficient for our Flow Matching model than Grooo--GooovvWasseeeset
F Toy example of goal-guided graph generation
We demonstrate the utility of our goal-guided framework of flow matching with a toy example, depicted Figure S2 (a) shows a trained unconditional flow matching model mapping noise distribution Po to data distribution P1- (b, c) illustrate the effect of temperature T on the exploration, with higher temperatures resulting in broader data point distribution. (d) shows how fine-tuning according to Equation [8 concentrates data in regions with higher rewards. (e-f) illustrate the corresponding flow matching trajectories.
G Additional Experiments Results
In this soction, we present additional metrics including Spectre (Spec.) and Validity&&NNvvoly&& Uniquess (Val.&Nov.. Uni.) across general graph datasets including the Planar and Enzymes datasets, as summarized in Tables S4 S5 and S6 The MMD kernel in the planar dataset followed the GruM (Jo et al. 2024). We also include the standard deviation of our results in Table S7 illustrating the consistency and superior performance of our method.
To further compare GGFlow with baseline models, we measured the MMD between the test datasets and of 1,024 generated graphs in the Ego-small and Commuity-small datascts. The results in Table SS demonstrate that GGFlow achieves the highest performance across all metrics, significantly outperforming other baseline models.
29

---

Published in Transactions on Machine Learning Research (12/2025)
(a)
(b)
(c)
(d)
Cenerated
Cenernied
Genertad
Generatd
-2-
22
-2
22
4-
(c)
(h)
-2
-2
-4
-8-
75 1000
Figure S2: (n-d) Data distribution of the flow matching model, To is the original distribution (orange), лi is the target data distribution (blue), and the red dots are the data distribution generated by the model. (e-h) In reinforcement learning, the flow matching model conducts exploration /sampiing trajectories
Table $4: Additional generation results on the generic graph datasets. Results are the means of different. runs. The best results are marked bold.
Ego-small
Community-small
Grid
Method
Step
Spec.
Nov.&Uni.
Spec.
Nov.&Uni.
Spec.
Nov.&Uni.
Training Set
0.006
30
0.012
100
0.009
GDSS
0.034
27.5
0.053
100.0
0.043
100.0
1000
GSDM
0.024
0.0
0.015
0.0
1000
DiGress
0.017
30.0
0.055
100.0
0.025
100.0
500
SwinGNN
0.016
52.5
0.025
55.0
0.008
100.0
500
GGFlow 0.006
32.5
0.031
100.0
0.022
100.0
500
Table S5: Generation results on the planar graph datasets. standard deviation.
The best results are marked bold. o denotes the
Planar
Method
Step
Deg.
Clus.
Orbit
Spec.
Val.&NNov.&&ue Unni.
Training Set
0.0002
0.0165
0.0002
0.0050
100
GDSS
0.0039
0.2593
0.1732
0.0370
0.0
1000
GRASP
0.0022
0.2749
0.0055
0.0098
0.0
200
DiGress
0.0003
0.0372
0.0098
0.0106
75.0
500
GruM
0.0004
0.0382
0.0095
0.0069
87.5
1000
GGFlow
0.0156
0.0196
0.0019
0.0091
97.5
500
0.0064
0.0037
0.0006
0.0012
2.5
30

---

Published in Transactions on Machine Learning Research (12/2025)
Table S6: Generation results on the Enzymes graph datasets. The best results are marked bold. a denotes the standard deviation.
Enzymes
Method
Step
Deg.
Clus.
Orbit
Avg.
Training Set
0.008
0.096
0.012
0.039
GraphRNN
0.017
0.043
0.021
0.043
GraphAF
1.669
1.283
0.266
1.073
GraphDF
1.503
1.061
0.202
0.922
GraphVAE
1.369
0.629
0.191
0.730
EDP-GNN
0.023
0.268
0.082
0.124
1000
GDSS
0.026
0.102
0.009
0.046
1000
GSDM
0.013
0.088
0.010
0.037
1000
DiGress
0.010
0.046
0.002
0.019
500
GGFlow
0.008
0.026
0.002
0.012
500
0.0041
0.0106
0.0008
0.0130
Table S7: Standard deviation and mean of generation results the general graph datasets. μ and denote the mean and standard deviation, respectively
Ego-small
Community-small
Grid
Metric
Deg-
Clus.
Orbit
Sepe.
Deg.
Clus.
Orbit
Sepc.
Deg.
Clus.
Orbit
Sepe.
0.005
0.033
0.005
0.008
0.011
0.030
0.002
0.031
0.030
0.000
0.016
0.022
0.007
0.012
0.003
0.001
0.006
0.012
0.002
0.002
0.008
0.000
0.003
0.001
Table S8: Generation results on the generic graph datasets with 1024 generated graphs. The best results are
marked bold.
Ego-small
Community-smal
Method
Step
Deg.
Clus.
Orbit
Sepc.
Deg.
Clus.
Orbit
Spec.
GraphRNN
0.040
0.050
0.060
0.030
0.010
0.010
GNF
0.010
0.030
0.001
0.120
0.150
0.200
EDP-GNN
0.010
0.025
0.003
0.006
0.127
0.018
1000
GDSS
0.023
0.020
0.005
0.047
0.029
0.068
0.004
0.151
1000
GSDM
0.003
0.008
0.0009
0.011
1000
DiGress
0.017
0.038
0.006
0.021
0.013
0.040
0.004
0.055
500
SwinGNN
0.004
0.023
0.003
0.023
0.003
0.088
0.010
0.016
500
GGFlow
0.004
0.004
0.0008
0.009
0.004
0.003
0.0006
0.018
500
Implementation Details
H.1 Algorithms of GGFlow
Details of the training procedure and guided training procedure are provided in Algorithm 3 and
4
H.2 Baselines Implementation
To benchmark the performance of GGFlow, we ensure consistency by using identical splits of training and test sets across all datasets. Below, provide the implementation details for each baseline model. To guarantee
31

---

Published in Transactions on Machine Learning Research (12/2025)
Algorithm Training Procedure of GGFlow
Require: Training set Strain {G = (V, EV, Pref.
1: for E {O..... Niter -1 ] do
2: t t u(0, 1), Sample G from Strain
3:
-V'VV, ~ Prof TGG G')
4:
// Sample from conditional probability flow.
Va (LO(V*, V ----)((1 Vo)) and E' (L0{E", B (1-
(1-1) -(L-
( G" GO) GraphEvo,, (G',t)
C= E P ((((:,1)=(( (Gt) (oo loyllogghn Pul (G aal1 = optimizer -pptiizer_update@...e
10: end for 11: 0" NN.er
12: return l"
Algorithm Training Procedure of Guided GGFlow by Reinforcement Learning
Require: 8o: 0, a, B, Noteps: traj, GO Pref, T, Nurain
1: 0 +- 0g
2: for E Nerain) do
AI 1/Nsteps
Collect flow trajectory (,, in traj.
for n E (0...., Nstops - 13 do
GG" (G), GraphEvo(G*,tt
Get GL+ by sampling from Equation 17 (VI+A), E++ -ttrt.
t=t+At
10:
Compute the reward function R(G++D ++ At).
Collect flow trajectory (g++ar, 1+ At R(G++ tt At)) in traj.
11:
12:
end for
13:
Update network using Equation 18
14:
t=0
15:
end for
16: return Guided flow matching model 0*
fair comparison, most bascline models are retrained three times, and the average results from these runs reported as the final outcomes in unconditional generation tasks. The results of the DeepGMG, GraphRNN and GNF for Ego-small and Community-small dataset are taken from their original papers.
GraphAF SShi et al.
We follow the implementation guidelines provided in the TorchDrug tutorials http:///g.../eooooootto////////a html)
2019
atttps:/ttorcdduugaaaa/ http://dwu...//ddoo//ttttttttoot
GraphDF (Shi et al.
2019
Model scripts are sourced from the DiG repository (https://github.comm
divelab/DIG/tree/ dig www.ttalleee... httpg//aaaaaaleeeaaaa////aaa///a
GraphVAE (Shi et al.
2019
Scripts are obtained from the GraphVAE section of the GraphRNN repository
htttps://github.ccom// com Jwwxuan http://ww....c..iaaaaaaaaaaaaaaa /ttpgggggggggggaaaaaoaareeeaeeee
MoFlow Zang & Wang 2020
Implementation scripts are taken from the MoFlow repository (https:
//github.com/ httpa//o.wwooaaaaaoo/oooooo/aaae
GraphEBM [Lu et al. 2021) We use the implementation available in the GraphEBM repository hhttps
wwwptthw...c.b.oooottee. AI AI/ Att.omddddA//rrrraaaaa
32

---

Published in Transactions on Machine Learning Research (12/2025)
EDP-GNN (Nu al. 2020) The model is implemented according to the scripts in the EDP-GNN
repository (https: / htt////gttttt github. com hoop///nonggrooorrppprrrrrriaaaa
GDSS [Jo et al. 2022b
Implementation details are sourced from the GDSS repository (https://github
com com/harxyjo97/CD33 GDSS
GSDM (Luo et al. 2023
Scripts are implemented from the GSDM repository (https://github.com
1tz0120/aast_Gren Generation Lainn_vi___pppe
PS- VAE (Kong et al.
2022
Implementation details are sourced from the PS-VAB repository (https:
SETTTPS-VALL
MolHF (Zhu et al.
2023)
The model is implemented according to the scripts in the MolHI repository
https://github. com ctt/
GRASP Minnllo al.
2024)
Implementation details are sourced from the GRASP repository (https:
//github.ccmm/ccaneee /RAAssssmmmm/RP GRASPP
Implementation details are sourced from the SwinGNN repository (https: //github. c LaSS--LLb/SwwGhh SwinCHN). The authors employ the "gaussian_ tt' MMD kernel, whereas other methods use ggaussian emd' or 'gaussian'. To ensure a fair comparison, we adopt the same kernel.
SwinGNN (aan al.
2023)
GruM (Jo et al. 2024
Scripts are implemented from the GruM repository (https://githubb.cmm
harryjo97/GruM,
DiGress Vignac et al.
2022
The implementation is based the DiGress repository (https://github
con/vvignaccDiGress)
H.3 Details of Generic Datasets
H.3.1 Dataset
Ego-small
This dataset consists of 200 small one-hop ego graphs derived from the Citeseer network (Sen
ct al. 2008 Each graph contains between 4 and 18 nodes.
Community-small This dataset includes 100 random community graphs, each formed by two communities of equal size generated using the E-R model (Erdös et al. 1960) with a probability parameter of p = 0.7. The graphs range in size from 12 20 nodes.
Enzymes The dataset comprises 587 protein graphs, with each graph representing the tertiary structure of enzymes sourced from the BRENDA database (Schomburg et al. 2004), which have between 10 and 125 nodes.
Grid
The dataset consists of 100 standard 2D grid graphs with 100 < |V| S 400.
Planar The dataset consists of 200 planar graphs, each with 64 nodes, generated using Delaunay triangula- tion on uniformly distributed random points.
Table S9: Statistics of the generic graph datasets
Dataset
type
Number of graphs
Number of nodes
Ego-small
Real
200
[4, 18]
Community
Synthetic
100
[12, 20
Enzymes
Real
587
[10, 125]
Planar
Synthetic
200
64
Grid Synthetic
100
[100,400]
33

---

Published in Transactions on Machine Learning Research (12/2025)
H.4 Details of Molecule Datasets
H.4.1 Dataset
QM9 It a subset the GDB-17 database and consists of 134,000 stable organic molecules, each containing up to 9 heavy atoms: includes 12 tasks related to quantum properties. We follow the train/test split from GDSS, using 12,000
carbon, oxygen, nitrogen, and fluorine (Ramakrishnan al., 2014p. The dataset
molecules for training and the remaining 1,000 for testing.
ZINC250k It contains 250,000 drug-like molecules with a maximum of 38 atoms per molecule (Iwwin et al., 2012). It includes 9 atom types and 3 edge types. For a fair comparison, we use the same train /test split as previous works, such as GDSS and GSDM.
Table S10: Statistics the molecular graph datasets
Dataset
type
Number graphs
Number of nodes
Number of node types
Number of edge types
QM9
Real
133,885
[1, 9]
3
2INC250k
Real
249,455
[6, 38
3
H.4.2 Metrics
For generic graph datasets, employ Maximum Mean Discrepancy (MMD) to assess the distributions of graph statistics, specifically degree distribution, clustering coefficient, the number of occurrences of 4-node orbits, and eigenvalues of the normalized graph Laplacian. In alignment with prior research (Jo et al. 2022b), we utilize specialized kernels for MMD calculations: the Gaussian Earth Mover's Distance (EMD) kernel for degree distribution and clustering coefficient, the Gaussian Total Variation (TV) kernel for eigenvalues of the normalized graph Laplacian, and a standard Gaussian kernel for the 4-node orbits. To ensure fair comparison, the size of the prediction set matches that of the test sct.
Validity We permit atoms to exhibit formal charges during valency checks because of the presence of formal charges in the training molecules. It is the fraction of valid molecules after valency correction or edge resampling.
Validity w/o correction This metric explicitly evaluates the quality of molecule generation before any correction phase, providing a baseline for raw generation performance.
FCD FCD quantifies the functional connectivity density within molecule by computing distances and connectivity between atoms, based on both structural and chemical features. It describes the three-dimensional structure, topological features, and chemical properties of molecules, making it valuable in fields such as drug design, compound screening, and molecular simulations.
NSPDK NSPDK assesses molecular similarity by comparing shortest paths within their graphical structures. It captures conncetivity patterns and chemical environments, effectively describing relationships and similarities between molecules. For two distributions p and y, the MMD using NSPDK is calculated
72
a (X,,
m
m
MMD & (p.9)
kNSTDK (V,)))))
(31)
n(n -
m(m - 1)
2
m
(x,)))(()))))
(32)
mi
Here, k-SPDK(.) denotes the NSPDK kernel function. X' the set molecules from distribution p. y the set of molocules from distribution q. and m represent the number samples drawn from distributions p and g, respectively. This formula quantifies the difference between the distributions p and q using the NSPDK kernel.
34

---

Published in Transactions on Machine Learning Research (12/2025)
H.5 Details of Conditional Generation
We included three guidance baselines in our conditional generation task:
DiGress model with guidance Utilizing the guidance method integrated into the DiGress model
Vignac
ct al. 2022
Direct supervised training (ST) involved selecting training samples from the dataset that satisfied
-- < 1.0 and retraining them using supervised learning settings identical those in Section 5.2
Supervised fine-tuning (SFT) This method involved fine-tuning pre-trained GGFlow model
molecules generated with / - n²1 < 1.0, maintaining the same training settings as in Section 5.2
These models were trained over 10,000 steps using the training settings detailed in Section 5.2 We then
generated 1,000 samples to calculate the results for each guidance method and the unconditional method, with the values of j estimated using Psi4 (Smith et al. 2020). We set the hyperparameters and / as 0.999 and 0.001.
Experiment Settings
1.1 Hyperparameter Settings
Table S presents the hyperparameters employed in our experimental setup. For each dataset, the final resutls in Table and Table 2 are the means of 5 different runs.
Table S11: Hyperparameter settings of different datasets
Hyperparameter
Ego-small
Community-small
Grid
Planar
Enzymes
QM9
ZINC250k
Number layers
5
5
9
9
Hidden dimension of X
256
256
256
256
256
256
128
Hidden dimension of E
128
128
128
128
128
128
64
Hidden dimension
128
128
128
128
128
128
64
Optimizer
Adamu
Adamw
Adamm
Adamw
Adamm
Adamw
Adammw
Learning
10
10-4
10
10
2 10
10-4
10
Batch size
64
128
4
64
512
128
Number of epochs
2000
3000
5000
5000
10000
1000
1000
Number sampling steps
500
500
500
500
500
500
500
1.2 Ablation Studies Settings
For the evaluation of varying inference steps, we followed the same experimental settings outlined in Scctions 5.1 and [.2] Samples were generated for 10 runs. The results were then visualized using the mean and variance across these 10 runs. It is important note that in DiGress, the number of inference steps is constrained by its predefined diffusion steps (N 500), the DiGress curve terminates at 500 inference steps.
For the ablation studies of GGFlow without Optimal Transport (GGFlow w/o OT), GGFlow without GraphEvo (GGFlow w/o Evo) and GGFlow without GraphEvo and optimal transport (GGFlow w/o both), we adhered the settings described in Sections 5.1 and 5.2 The final results were obtained by averaging the outcomes from five different runs.
Due to the excessive time required to compute metrics for 10,000 generated molecules, we conducted the experiments on inference steps and model stability using smaller set of 1,000 generated molecules to calculate the test. metrics on ZINC250k dataset.
To further investigate the advantages of optimal transport, we present generation results with varying inference steps on the Community-sni dataset. As shown in Figure S3 GGFlow demonstrates superior generation quality compared to GGFlow (w/o OT), exhibiting narrower confidence intervals and comparable performance
35

---

Published in Transactions on Machine Learning Research (12/2025)
with fewer inference steps, which suggests that optimal transport enhances sampling both efficiency and stability.
GGFlow (w/o OT)
GGFlow
0.04
0.03
Average
0.02
0.01
0
200
400
600
800
1000
Inference Step
Figure S3: Ablation studics of varying inference steps on Community-small datasct
0.150
GGFlow w/o Both
GGFlow w/o OT
0.145
Edge Cross Entropy
0.140
0.135
0.130
0
100
200
300
400
500
Epoch
Figure $4: Cross-entropy of edges the validation set for each epoch on ZINC250k dataset
36

---

Published in Transactions on Machine Learning Research (12/2025)
1.3 Generation Times with Different Graph Scales
We conducted experiments to evaluate generation times across various graph scales using the Euzymes dataset. The GvaphEvo model was configured with six layers, and generation time was measured for a single graph on an NVIDIA A100 80G GPU.
Table S12: Generation times with different graph scales.
Number of Nodes
10
50
100
200
400
Time (s)
1.92
4.24
12.15
48.59
235.8
Visualization
37

---

Published in Transactions on Machine Learning Research (12/2025)
Community Small
Grid
QM9
90
ZINC250K
Butt
Figure 85: Visualization of generated samples of our model in different. datasets
38


Reviews:
[{"review_id": "F2yfKSMMyk", "reviewer": "Reviewer_NEVR", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- Strong empirical performance: GGFlow achieves competitive or state-of-the-art results on multiple benchmarks for both unconditional and conditional graph generation tasks.\n\n- Generalizability: the model adapts well to various graph domains, including molecule and generic graph generation.\n\n- Thorough ablations: ablation studies convincingly demonstrate the contributions of flow matching, optimal transport, and the GraphEvo component.\n\n## Weaknesses\n### Motivation needs to be improved\nThe transition from diffusion to flow matching lacks a strong theoretical or empirical justification. The paper motivates GGFlow by stating that diffusion-based models suffer from training instability and inefficient sampling. However, these claims are overstated and not convincingly supported. Many modern diffusion models (e.g., DDPM, DDIM, NCSN++) are known for stable training and can generate high-quality samples with as few as 20–100 inference steps. In contrast, GGFlow still requires 500 steps, limiting the claimed efficiency advantage.\n\n### Unsubstantiated claims about training instability\nBeyond the issues noted above, the paper asserts instability in diffusion training without providing empirical or theoretical evidence. Figure 4, which is meant to support this point, does not clearly show faster convergence. While GGFlow variants achieve lower training loss compared to DiGress, the convergence speed — i.e., the rate at which loss stabilizes — appears similar. Moreover, the lower loss may simply reflect the use of a more expressive architecture (e.g., GraphEvo) rather than an inherently superior training dynamic. Without test loss or generalization metrics, it’s unclear whether this indicates meaningful improvement or potential overfitting.\n\nFigure 4 does not clearly demonstrate faster convergence as claimed. While GGFlow variants achieve lower training loss compared to DiGress, the convergence speed — i.e., the rate at which the loss stabilizes — appears similar. Moreover, since the training objective is shared, the lower loss may reflect higher model capacity (e.g., due to the more complex GraphEvo architecture) rather than inherent advantages of the flow matching framework. Without accompanying test losses or generalization metrics, it is unclear whether this reflects meaningful improvements or potential overfitting.\n\n### \tLimited sampling efficiency advantage\nAlthough the paper emphasizes improved sampling efficiency, GGFlow still requires 500 inference steps — which is considerably higher than diffusion models using DDIM or ODE-based samplers, which often require only 20–100 steps to generate high-quality outputs.\n\n### Inaccurate classification of prior methods\nUnless I am mistaken, the paper inaccurately categorizes VAEs and GANs as autoregressive models, which is conceptually incorrect. This misclassification reflects a lack of precision in the exposition and may lead to confusion, particularly for readers less familiar with the distinctions between generative model classes.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces GGFlow, a generative model for graphs based on discrete flow matching with efficient optimal transport. The core idea is to replace stochastic diffusion processes with deterministic flow matching, aiming to improve sampling efficiency and training stability. GGFlow integrates an edge-augmented graph transformer, GraphEvo, to better capture node and edge interactions, and introduces a goal-guided generation framework using reinforcement learning for conditional generation. The method is evaluated on both generic graph and molecule generation tasks, demonstrating state-of-the-art performance across multiple benchmarks. Empirical results and ablation studies suggest that GGFlow performs well with fewer inference steps than traditional diffusion-based approaches, although the model still requires 500 steps.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Clarify and refine the motivation\nThe current framing overstates the limitations of diffusion-based models, particularly with respect to training instability and sampling inefficiency. Section 1 (Introduction) should be revised to better motivate the transition to flow matching, ideally with more accurate claims and clearer justification. The authors should either provide empirical evidence to support their critiques of diffusion models or revise the narrative to offer a more balanced and accurate comparison with recent developments (e.g., DDIM, ODE-based sampling, consistency models).\n\n### Justify claims about convergence and training stability\nFigure 4 is used to support claims of faster convergence, but it only shows lower training loss, not faster stabilization. Including test loss curves or validation metrics would help determine whether the observed improvements reflect better generalization or just overfitting. \n\n### Polish presentation and terminology\nFix technical inaccuracies (e.g., model classifications) and clarify explanations to ensure the text is precise and accessible.\n\n### Minor comments and typos\n- After equation (4), $ReLU(a)$ is defined but never used.\n- Consider renaming Section 4 to “Experiments” for clarity.\n- Typo: 4.2. Moleucle -> 4.2. Molecule.", "broader_impact_concerns": "I do not have any specific broader impact concerns regarding this submission. The paper focuses on a methodological contribution to generative modeling for graphs, and no significant ethical, societal, or misuse-related risks are apparent from the content as presented.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Yp2oFeeN42", "reviewer": "Reviewer_rbbG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The integration of discrete flow matching with optimal transport for graph generation is a valuable contribution to the field.\n2. The paper includes extensive experiments across multiple datasets and metrics, demonstrating improvements over existing methods.\n\nWeaknesses:\n1. While computationally efficient, the choice of Hamming distance as the metric for optimal transport lacks sufficient theoretical justification, particularly regarding its appropriateness for capturing graph structural similarities.\n2. The authors inadequately address the implications of limiting invariance to identical permutations; a more thorough discussion of how this constraint affects real-world applications is needed.\n3. As this work combines multiple existing techniques, the ablation study requires substantial expansion:\n    - Experiments should be conducted across multiple datasets to ensure generalizability.\n    - The specific contribution of the RL-based fine-tuning mechanism is not isolated from other components.\n4. The authors claim that \"the integration of Optimal Transport significantly boosts sampling efficiency,\" yet Figure 3 shows minimal separation between GGFlow and GGFlow(w/o OT) curves, contradicting this assertion.\n5. The concept of \"training stability\" referenced in Figure 4 lacks a clear definition and quantitative metrics, making it difficult to objectively evaluate this claimed benefit.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces GGFlow, a discrete flow matching generative model for graph-structured data.\nIt has three key designs:\n- Discrete flow matching with mini-batch optimal transport for graph generation.\n- Edge-augmented graph transformer called GraphEvo.\n- Reinforcement learning-based goal-guided framework for conditional generation.\n\nThe authors demonstrate that GGFlow outperforms existing methods on both unconditional and conditional graph generation tasks, showing superior performance on several benchmark datasets, including generic graphs and molecular structures.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Refer to weakness (critical to securing my recommendation for acceptance).", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "U7wzwN1VN3", "reviewer": "Reviewer_duAs", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n1. The paper introduces new modules to improve discrete flow matching for graph generation. The approach shows empirical improvements over existing approaches.\n\n**Weaknesses**\n While a number of components have been introduced there is a lack of discussion on the design decisions and the actual contributions of the paper. For example, the addition of optimal transport to discrete flow matching model is stated as the first contribution but the claimed efficient optimal transport is explained in a. short paragraph in section 3.2. It is not clear what exactly is done to make it efficient for graphs and how does it lead to improvement in sampling efficiency and training stability.\nSimilarly the design decisions for the other two components are missing.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper develops an approach called GGFlow which is a discrete flow matching generative model for graphs. To improve generative performance the authors augments the approach with optimal transport and an edge a-augmented graph transformer as well as a goal guided framework for conditional generation. Empirical evidence is provided for improved generation over the baselines.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please explain the design decisions behind each component. Specifically explain how optimal transport is adapted for graph generation, why is it efficient and how does it aid to training stability and sampling efficiency.  As the work is developed on existing discrete flow matching models, please motivate/ elaborate on any non-trivial adaptations that are made to make specific components work for graphs.\n\n 2. I found another related work which also seems to focus on sampling inefficiency as the current work : https://arxiv.org/abs/2410.04263. I would request the authors to add a discussion on how the current work is different from DeFoG.", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)