Global and Local Topology-Aware Graph Generation via Dual Conditioning Diffusion
Authors: Yuhang Xie
Sinno Jialin Pan
Venue: ICLR 2026 Poster (2026)
Keywords: generative model; AI for science; conditioning method
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Graph generation plays an important role in various domains such as molecular design, protein prediction, and drug discovery. However, generating graph-structured data poses challenges due to the complex dependencies inherent in graphs, spanning from intricate local substructures to broad global topologies.  Although recent advances in graph-generative models have made notable progress, traditional node-level generative paradigms may have difficulty simultaneously capturing the multiscale dependencies in graphs. To address these challenges, we propose a unified latent diffusion model that jointly learns local and global topological information, enabling effective and efficient graph generation. Besides, our approach introduces a dual conditioning mechanism designed to promote dynamic interaction between local and global information, equipping the generative model with global and local awareness to better capture the coupled dependencies within graphs. Our method can largely promote the joint modeling of global and local information and substantially improve the quality of the generated graphs. Extensive experiments consistently demonstrate the effectiveness of our method.

Published as a conference paper at ICLR 2026
GLOBAL AND LOCAL TOPOLOGY-AWARE GRAPH GENERATION VIA DUAL CONDITIONING DIFFUSION
Yuhang Xie
Sinno Jialin Pan
Department of Computer Science and Engineering The Chinese University of Hong Kong, Hong Kong yuhangxied ink. www.cuckkkooo edu. hk, sinnopan@cuhk. edu
ABSTRACT
Graph generation plays important role in various domains such molecu- lar design, protein prediction, and drug discovery. However, generating graph- structured data poses challenges due the complex dependencies inherent in graphs, spanning from intricate local substructures to broad global topologies. Al- though recent advances in graph-generative models have made notable progress, traditional node-level generative paradigms may have difficulty simultaneously capturing the multiscale dependencies in graphs. To address these challenges, we propose a unified latent diffusion model that jointly learns local and global topo- logical information, enabling effective and efficient graph generation. Besides, our approach introduces a dual conditioning mechanism designed to promote dy- namic interaction between local and global information, equipping the generative model with global and local awareness to better capture the coupled dependencies within graphs. Our method can largely promote the joint modeling of global and local information and substantially improve the quality of the generated graphs. Extensive experiments consistently demonstrate the effectiveness of our method.
INTRODUCTION
Graph-structural data provide powerful representations for describing objects and their relationships in diverse real-world domains, such as social networks (Singh et al., 2024), biological systems (Liu
al., 2024 Zang Wang 2020), and traffic networks (Cui et al. 2019). Among the core tasks in graph machine learning, graph generation has emerged as powerful tool with significant potential
in diverse applications such as molecular structure modeling (Xu et al. 2023 Vignac et al. 2022
Luo al. 2023), circuit design (Shahane et al. 2023) and source code generation (Dai al. 2018
Currently, advances in deep models, including generative adversarial networks (Goodfellow et al, 2020), diffusion- biffusiio-bbsed models (Ho et al. 2020: Jo et al. 2022 Luo et al., 2023). and flow- blswd methods (Chen et al. 20188] Bengio et al.
2023 Zhang et al. 2024b), have paved the way for ef- fective graph generation. Unlike densely distributed image data, graph data can be extremely sparse and often exhibits complex topological information, especially the coupled local and global depen- dencies within distinct structures (Guo & Zhao 2022). For instance, social networks exhibit intri- cate dependencies both within individual communities and between different communities (Singh al. 2024); likewise, molecular graphs demonstrate complex relationships both within and among substructures (Zhao et al.) 2012). Specifically, the local structure of molecule, such as the specific functional groups plays crucial role determining its chemical reactivity. However, the global formation, including the overall structural topology and the spatial distribution of functional groups, also significantly impacts its overall properties and functional performance (Livingstone) 20000) Therefore, the intrinsic and distinct dependencies present in graphs highlight the importance of in- tegrating both global and local information to enable a more precise graph generation process.
Recently, some effective models have incorporated global information to enhance the graph genera- tion process. For instance, SubgDiff (Zhang et al. 2024a) introduces a subgraph prediction module that integrates substructure information into diffusion models, thereby improving molecular repre- sentation learning. Similarly, Graphusion YYngg et al. 2024) leverages graph-level useudo-labels derived from clustering algorithms to provide informative guidance during the generation process.

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However, these approaches may overlook the dependencies among substructures and have limita- tions in modeling the joint distribution of global and local information. This raises a natural and important question: How can we design a unified generative model that dynamically capture both global and local topological information to enable effective graph generation?
To answer this question, propose unified dual conditioning latent diffusion model, termed DualDiff, which is capable of effectively modeling the joint distribution of global and local in- formation, facilitating robust and stable generation process. Concretely, DualDiff first maps the original graph space and diverse graph features into a unified latent space via a pretrained graph autoencoder, then extracts the global information according to the local representations, Subse- quently, DualDiff employs a two-branch diffusion process to learn topological dependencies at both the node and subgraph levels within a unified framework. To further advance the joint modeling of global and local information, a dual conditioning mechanism is introduced to promote interac- tion between these two branches, wherein global and local information are alternately utilized as conditions to guide the learning process of the complementary branch.
By leveraging the two-branch diffusion process and dual conditioning, DualDiff can be equipped with both global and local topological awareness, enabling it to dynamically capture the dependen- cies within the coarse searse-rrained global structures and the fine-grained local details. Experimentally, conduct comprehensive evaluations graph generation benchmarks, further demonstrating the effectiveness of our proposed method. Our main contributions are summarized as follows:
We highlight the limitations of conventional graph generative models in capturing the joint distribution of global and local topological information within graphs. To this end, we propose DualDiff, an effective latent diffusion-based generative model capable of simulta- neously learning the dependencies inherent in both global and local structures.
By incorporating the proposed two-branch diffusion process and dual conditioning mecha- nism, DualDiff can effectively capture the joint global and local topological dependencies.
Extensive experiments across diverse benchmarks, including generic graphs and real- world molecular datasets, demonstrate that DualDiff consistently outperforms existing methods, largely illustrating the effectiveness of our proposed method.
2 RELATED WORK
Generative Models on Graph. Graph generative models have been widely adopted across var- ious domains. Early approaches include graph-based variants of variational autoencoders (VAEs) Kingma et al. 2013; Kipf & Welling 2016 Jin et al.] 2018) and generative adversarial networks
(GANs) (Goodfellow et al. 2020 De Cao & Kipt 2018 Krawezuk et al. 2020), flow-based mod-
els (Lippe & Gavves 2021; Luo et al. 2021), and autoregressive models Yoou et al., 2018 Liao
ctt al. 2019). More recently, diffusion- based models have demonstrated state-of-the-an performance
on graph generation tasks (Jo et al. 2022, Qiang et al. 2023] Yan et al.) 2023 Luo et al. 2023).
Despite these advancements, effectively modeling the complex and coupled topological dependen- cies inherent in graphs remains significant challenge. To address this limitation, we propose unified framework, DualDiff, designed to simultaneously capture both global and local topological information, thereby facilitating a more efficient and expressive graph generation process.
Latent Graph Diffusion Model. Latent diffusion model (LDM) (Rombach al.) 2022 Podell et al. 2023) leverages the idea of performing the diffusion process in a lover-dimensional and com- pact latent space instead of the original data space, which greatly improves efficiency and scalability. Building on the success of LDM in images, GEOLDM (Xu et al. 2023), EDM-SYČO (Ketata et al.) 2025). and LGD (Zhou et al., 2024) extend LDM to molecular and graph generation tasks. They embed the graph structures and features into a latent space by a pretrained autoencoder and conduct a diffusion process in the latent space, exhibiting competitive performance and high computational efficiency. However, these methods still rely node-level generative methods, which treat each node as an independent entity and may overlook dependency within substructures.
Conditioning on Diffusion Model. Conditioning (Ho al. 2022 Nichol & Dhariwal 2021 is a widely used technique for guiding diffusion models to generate samples in a targeted and mean- ingful manner. Traditional conditioning methods typically incorporate external information, such as
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class labels auxiliary inputs like low-resolution images (Zhang et al. 2023 More recently, self- conditioning (Chen et al. 2023) allows models to leverage their own previous predictions. resulting in notable improvements in sample quality. In addition, GPrinFlowNet (Mo et al. 2024) intro- duces low-to-high frequency generation curriculum, effectively preserving semantic information and achieving high-quality conditional graph generation. Distinet from these prior approaches, we propose a dual conditioning mechanism that facilitates interaction between global and local infor- mation. This design endows our model with both global and local topological awareness, enabling more effective learning of the diverse dependencies present in graph-structured data.
3 PRELIMINARY
Diffusion Models. Diffusion models have demonstrated strong capabilities in modeling complex data distributions (Sohl- Dicks-icckteen al. 2015 Ho al. 2020). In these models, the forward process gradually corrupts a data sample Xn by adding Gaussian noise over T steps, producing sequence X, according to 7(x; xx N (x+; vT -, B,T,, )) where B. is the noise schedule. The objective
of diffusion is to learn the reverse process pa xx-1 X+). which denoises X, to recover the original distribution. Elucidated Diffusion Models (EDMs) (Karras al. 2022) extend this framework by generalizing the noise schedule and reverse process parameterization. In EDM, the model is trained to predict clean data from noisy inputs across different noise levels. Specifically, given a noise level a, the denoising loss is defined as Pa (x)x) Do σ) -x| where Do (x, o) is the denoising network, and X is the noisy data sampled as pr (x[x) = N(;; x, T) During reverse process, the learned denoising network approximates the score function Vx log p(x; σ = (x, σ) x)/o²,
enabling sample generation via numerical solvers described in (Song al. 202
CiIdiiniiiiinnnn. Self-conditioning is a tool to improve diffusion models by allowing the denois- ing network to utilize its own previous predictions as conditions (Chen et al. (023). Specifically, seff-conditioning feeds the previously predicted clean sample Xn back into the model as an additional input, which can largely enhance sample quality and consistency with minimal computational over- head. During training. self-conditioning is applied with a probability to prevent the model from over-relying on self-conditioned inputs, thereby improving the robustness of the learning process.
METHODOLOGY
Problem Definition. In this paper, consider graph generation from scratch. Let N denote the number of nodes in a given graph, then a graph can be defined as 9 = A), where H E IR xd, denotes the node feature matrix, and A E N represents the adjacency matrix. For tasks related to biology and chemistry, the 3D coordinate matrix X E R xx3 can also be available, The graph autoencoder utilized in our latent diffusion framework consists of an encoder & and decoder Dio- The encoder &g transforms G into latent representation Z Nxd We Z₁ R. to denote the local information and Zg I Kxdd to represent the global information, where Zg is obtained by applying clustering to Z₁ and K is the number of clusters. By employing advanced graph clustering methods, Zg can effectively encode important global information within graphs. Our objective is to employ a unified latent diffusion model to model the joint distribution pl Zi, Zg).
Overview. The primary motivation behind DualDiff to endow the model with both global and local structural awareness. For example, in molecular generation tasks, DualDiff can accurately cap- ture local details, such as specific functional groups or molecular fragments, while simultaneously considering the molecule's overall topology and the distribution of substructures. The architecture of DualDiff is depicted in Figure T We first describe the construction of the autoencoder and global information extraction in Section 4.1 Then, the details of Difl will be illustrated in Section 4.2
Finally, we summarize the training and sampling scheme of DualDiff in Scction 4.3
4.1 AUTOENCODER CONSTRUCTION AND GLOBAL INFORMATION EXTRACTION
AutoEncoder. In graph-structured data, there often exist many meaningful features, such as molecular conformation positions and charges. To fully utilize these informative features, we lever- age the Latent Graph Generation (L.GD) paradigms map graphs and the features into unified
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Procese (i): Global - Local
Ply
Zry
Au
Gruph/ Mouacate
Encoder
GNN
Dusl Conditionnna
Dus Condioinin
prlor distribution
Liner
Giebal
Agu
2
Linear
Linear
Dg
Deccdom
Process [il]: Local Globol
busl Condiroring
Dual Condttowing
Ags
GNN
Generated Graph// Hoteouto
Original Space
Pooling
Linear
Api
Pay
Reverse Process
Dual Conditioning Mechanism
Figure I:
The workflow of DualDiff (Left) and details of the dual conditioning mechanism (Right).
latent space, thereby integrating rich feature information to facilitate generation. Specifically, follow the settings in previous work [Rombach et al., [022]] Xu et al. 2023 Ketata et al. (225)) and pretrain an autoencoder to enable the transformation between graph space and latent space. The en- coding and decoding processes can be formulated by 9 ( = (G) auI) and pe (g) Zi. Zg).
Regarding the encoder architecture, in chemical and biological domains. nodes are generally en- dowed with 3D coordinate information, which inherently encodes spatial adjacency relationships. encoder with
To exploit this property, we incorporate equivariance into & by parameterizing Equivariant Graph Neural Networks (EGNNs) (Satorras et al.) (021). thereby ensuring that the in- duced latent representations are SE(3)-equivariant For general graph-structured data lacking ex- plicit coordinate information, Eg can be instantiated with various established graph neural network architectures, such as GIN (Xu et al. 2019). GCN (Kipf & Welling 2017). and GAT (Veličkovid et al., 2018), to effectively facilitate message passing among nodes. To reconstruct the original graph, our decoder exploits both global and local structural information, which can be implemented lightweight MIPs. The overall framework is summarized as follows:
Z Z Tol Zg = GlobalExtraction/ Zz = D.. Zg).
Z = Eg()
where GlobalExtraction represents the operation of global information extraction, which will be explained later. Then, the whole framework can be effectively optimized by:
Cree = (gs)) (2)) (g Z,, Zy)]
In practice, the reconstruction loss can be calculated as the L2 norm or the cross-entropy function. More details of the autoencoder and latent space can be found in Appendix A.I
Global Information Extraction. Recent studies have demonstrated the critical role of global in- formation in graph representation learning, particularly in applications like molecular property pre- diction and social network analysis (Yang et al. 2019: Fang et al.) (022). By incorporating global information, models can capture long-range dependencies and holistic structural patterns that cannot be fully represented by local features, thus largely improving the representation capability.
To extract global topological information from a graph, we leverage graph clustering methods (Scha- effer 2007). which are powerful tools for uncovering intrinsic structural patterns within graphs. To obtain topologicall--enhanee global information, we first employ different clustering strategies based on the nature of the graph. Specifically, for molecular graphs, apply the K-means algo- rithm in the atom coordinate space to generate coometry-enhanced class labels. For generic graphs, utilize spectral clustering (Von Luxburg 2007) that partitions nodes according to the eigenvectors of the graph Laplacian matrix. Following the clustering process, we aggregate node features within each cluster using pooling operation to derive cluster-level embeddings. The overall process of

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global information extraction can be formulated follows:
9 = CChstering(G)
Zg = Pooling(Sy. Z) E RRxdd
(3)
where S E {0, 1 N the affiliation matrix indicating the cluster assignment of each node. By introducing lopologically-enhnnee global information, our model can effectively capture dependen- cies among global structures and guide the generation of local details more efficiently. We provide more detailed explanations regarding the effectiveness of the clustering methods in Appendix C.8
4.2 TWO-BRANCH DIFFUSION WITH DUAL CONDITIONING
To integrate global and local information, we propose a unified diffusion framework that explicitly models both node-level and cluster-level dependencies. Furthermore, dual conditioning mecha- nism proposed to facilitate effective information exchange between global and local information, thereby promoting the model to capture the joint distribution of global and local representations.
Two-Branch Diffusion on Global and Local Information. Given that global structures and local details within a graph generally exhibit distinct topological characteristics, we introduce a unified two-branch diffusion process operating at both the node and cluster levels, which enables the model to effectively capture local and global information, respectively. Following (Song et al. (021). the forward process of this diffusion can be formulated as system of stochastic differential equations (SDEs) defined in the latent space, as follows:
dZ = fi, dt + St.dW
(4)
dZz. = fg.0 Z ) + dw g.t.
where ft,(()) and f.,(()) denote the drift coefficients, S1, and $9.8 are the diffusion coefficients, which are typically formulated deterministic functions of time Wi, and W g. represent the standard Wiener processes. Correspondingly, the reverse-time SDE system of (4) can given by:
dZ (fia(Z),)) - .t. Z log (Z..) dt + st.dWW
(5)
dZ g. (fg,((Zg.)) a.. V Vz, log Pi Z)):) di + sy. g,bs
where di dtt is the negative infinitesimal time step, Wt. and W g.t represent the reverse-time Wiener processes, and V log p.() is the score function. Under the EDM framework, we can set fie (Z,,)) = 0, fg.r( = O, and 81, = *g.t = 22. To predict the clean global and local latent embeddings, employ separate denoising networks, Do, and Do,- which be instantiated as GNNs in practice. Then, the overall training objective can be formulated as:
E E2 (2,00:2,0)000 (2. 2, 2g Do. (Zi, o) Z + [|D (Zg. Zg.o
(6)
In the proposed two-branch diffusion process, the denoising networks designed to capture node- level and cluster-level dependencies independently. However, this architecture has limitations in modeling the joint distribution between global and local information. To further promote dynamic information exchange and enable joint modeling of global and local representations, we introduce a dual conditioning mechanism, described in the following part.
Dual Conditioning Mechanism. In this part, we propose a dual conditioning mechanism to achieve dynamic interactions between different topological knowledge, thereby equipping the model with comprehensive global and local topological awareness. According to the conditional probabil- ity formulation, the joint distribution of global and local information can be expressed as:
p(Z, Z,) = p( Zi Zz)))) = p(Zgl (Zt).
This equality implies that modeling the joint distribution of global and local information can be decomposed two complementary processes: (i) introducing global information into the modeling of local details, associated with p( [2 22 and (ii) incorporating local information into the modeling of global structures, associated with p( Zg Zi).
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Specifically, we first follow the self-conditioning and obtain the previously estimated Zí,0 and Zo0 produced by Do, and Da respectively. Then, we combine local and global information as condi- tions to alternately guide the local and global diffusion process. Let C₁ and Cg respectively denote the conditions introduced to the local and global diffusion processes, then they can be given by:
- ((2,,) 200)) 0), with prob, p. related to p( Z 2 )p( 2g) (0, (Z,,0 2 g,,)), with prob. 1 1-p. related to p( ( Z
(CI,C,)
(8)
where p denotes the frequency of process (i). In detail, during process (i), to match the first decom- position pl Zi) Zg )p(())((, (3)) in 7 we use C (Z..o. Zg.0) to incorporate both self-condition and global information to guide the learning of Z, thereby modeling p( Zi| Meanwhile, C. is set to 0 to align with p( g)) Similarly, we can obtain (C), Cg) (0, (Z Zg,0)) during process (ii). More interestingly, this strategy naturally matches the self-conditioning. where the condition is set to zero with a certain probability to enhance the robustness of the generation process, further guaran- teeing the effectiveness of the proposed dual conditioning. A comparison of different conditioning methods is provided in Figure 2 Next, we will present detailed description of the two processes used in the dual conditioning mechanism, as illustrated in the right part of Figure
(i) Incorporating global information into the local structure. According to the conditional probability D we can introduce the global information promote the generation process of lo- cal topologies. Concretely, inspired by FiLM (Perez et al., 2018), we first pass the predicted global representation 2g.0 through linear transformation to yield scale and shift parameters, y² I and 3 E H (i 1, l...., K). Next, each node is assigned to cluster 1/s based on the similarity between Zr, and Zy.0- Finally, the scale and shift parameters associated with each cluster are utilized to steer the distribu- tion of the final output Zi,- thereby integrating global awareness into the modeling of local struc- tures. Formally, have:
21,+1
Z1,+1
21
ggt+1
2,,0
1,0
2gg
7
Z1,5
XLC
Agh
%
9.0
R1,--
21A-1
27.0
21,0
21,0
2g
(b)
(c)
Figure 2: Comparison between different conditioning methods during the reverse process. (a) diffusion without
conditioning; (b) self-conditioning; (c) dual conditioning.
2/00 7 @ Zi,0 + 3%; where yı = arg max sim Zi,0 20,0); 21,0 = GNNGNNNNNN 20,0 oz). (9)
j=l....,K
(ii) Incorporating local information into the global topology. In addition process (i), local infor- mation can also be leveraged to help predict global structures, thus modeling p( Z Specifically, the estimated local details Žι,0 can be processed through message passing (MP) and pooling (Pool) operations to obtain a low-dimensional global condition C. This condition is then concatenated with the self-condition 2,00 as the final condition of the denoising network. Formally, we have:
C = Linear(Pool(MI (Z,,))) + C = Concat g.0, C*) + Z = GNN g.t. C,o.). (10)
In conclusion, the dual conditioning mechanism can naturally integrate processes (i) and (ii), en- abling dynamic interaction between global and local information and equipping DualDiff with global and local topological awareness, which largely promotes the effectiveness of the generation process. We also include more details of the dual conditioning mechanism in Appendix A.4
4.3 TRAINING AND SAMPLING
In this section, we present the detailed training and sampling procedures for DualDiff. Consistent with the latent diffusion model (LDM) framework, we employ two-stage training strategy: an au- toencoder is first pretrained, and remains fixed throughout both the diffusion training and sampling phases. To facilitate more stable sampling process. draw inspiration from the server-client communication paradigm, where the server updates global parameters after accumulating several updates from clients to promote training stability (McMahan et al. 2017). We analogize the global

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clusters to the role of the global server and their corresponding nodes to the local clients. Cor- respondingly, process (i) and process (ii) can be viewed as local update and global aggregation, respectively. Therefore, enhance the stability of the sampling process. Besides, this alternating strategy can significantly improve sampling stability and model performance, compared to executing two processes simul- lancously. Further details regarding the effectiveness of dual conditioning and the corresponding theoretical insights can be found in Appendix (D] Finally, with the integration of dual conditioning, the overall loss objectives are updated as follows:
alternate m steps of process (i) with single step of process (ii) to
B (Z (19)), 22,1,, Zo.0) De, [z, Ci Z1,ol =,,03 (Z,: -
(11)
Overall, the training and sampling process can be shown in Algorithm [ and Algorithm 2
Algorithm Training Algorithm DualDiff
Algorithm 2 Sampling Algorithm DualDiff
Input: input graph G, denoising networks Dø, and Dag. autoencoder & and Dy. noise sched- uler {ot). diffusion steps T, training epochs E. Output: learned Do, and Do
Input: trained Do. and Do.- trained decoder D., noise scheduler {az}, sampling steps T.
Output: generated graph G.
1: Zi, Z prior
1: Pretrain &g and De by optimizing 2 and
2: for t = T 1 to do
then fix the parameters.
3:
if t % (m + 1)
then
2: Zı Zg,0 ~ qa (-19)
(Ci, C.)) = (0, (Z Zg.0)) 5: else (C,, = ((Z 2 9000)
3: for = 0 E 8--1 do
4:
(0,
5:
Zr.t; Zy.t Po, (Zi, Zg Zi,u, g.0)
6:
Zi,- = solver Do. (Zi++l, 0r,0)
6:
Obtain Ci and Cg by (8)
7:
H solver De, g Cg Ø++1))
7: Update Da, and Do, by (II)
8: end for
8: end for
9: = Pel ZZ Zg.0)
9: Return: Do. Do,
10: Return:
EXPERIMENT
Experimentally, we conduct comprehensive evaluations our proposed DualDiff framework on two widely adopted graph generation tasks: generic graph generation and molecular generation.
5.1 EXPERIMENTAL SETUP
Following previous methods, we leverage eight graph datasets to evaluate our proposed DualDiff extensively. For generic graph generation tasks, we include Ego-small, Community-small Grid, Planar, and SBM datasets. For molecular graphs, involve mainstream ZINC250K, QM9, and MOSES datasets. For a fair comparison, we follow the experimental and evaluation setting of (Liu et al. 2025 Jo et al. 2022 Vignac al. (002) with the same train/test split. More details and descriptions of datasets can be found in Appendix B.2
Datasets.
Baselines. We compare DualDiff with competitive generative models. Specifically, for generic graph generation, choose competitive GraphRNN (You et al. 2018), GDSS (Jo et al.) 2022 DiGress (Vignac et al. 20222. GSDM (Luo et al. 2023), Graphusion (Yang et al. 2024). GruM (Jo
et al, 2023), GBD (Liu et al. 2025), and GraphBFN (Song et al. 2025a). For molecular generation,
we additionally select advanced MoLeR (Maziarz et al. 2021 GraphArm (Kong et al.
2023
MicCaM (Kong al.] 2023), MAGNet (Geng al. 2023
SwinGNN (Yan et al. 2023
LGD
(Zhou et al. 2024), and EDM-SyCo (Ketata et al. 2025).
Evaluation Metrics. Following (You et al. 2018 Du et al. 2021 Jo et al. 2022), for generic
graph generation, we use the maximum mean discrepancy (MMD) to compare the distributions of statistics between the same number of generated and test graphs, such as distributions of degree (Deg.). clustering coefficient (Clus.), the number of occurrences of orbits with 4 nodes (Orbit), and the cigenvalues of the graph Laplacian (Spec.). In addition, we report V.U.N. scores, representing the percentages of valid, unique, and novel graphs, to evaluate how well the model captures both

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intrinsic characteristics and global graph properties. For molecular generation, follow the met- rics used in Guacamol (Brown et al. 2019), and report scores of KL divergence (KL), the Fréchet ChemNet Distance (FCD) (Preuer al. 2018). Novelty, Uniqueness, and Validity.
Table 1: Comparison advanced models on Planar and SBM datasets. More experiments on the Ego-Small, Community-small, and Grid datasets are included in Appendix C.I
Datasets
Planar
SBM
Metrics
Deg.
Clus.,
Orbit,
Spec.
V.U.N.
Deg.l
Clus.
Orbitl
Spec.
V.U.N.1
Training sel
0.0002
0.0310
0.0005
0.0052
100.0
0.0008
0.0332
0.0255
0.0063
100.0
GraphRNN
0.0049
0.2779
1.2543
0.0459
0.0
0.0055
0.0584
0.0785
0.0065
5.0
GDSS
0.0041
0.2676
0.1720
0,0370
0.0
0.0212
0.0646
0.0894
0.0128
5.0
DiGress
0.0003
0.0372
0.0009
0.0106
75.0
0.0013
0.0498
0.0434
0.0400
74.0
GBD
0.0003
0.0353
0.0135
0.0059
92.5
0.0013
0.0493
0.0043
0.0047
75.0
GruM
0.0005
0.0353
0.0009
0.0062
90.0
0.0007
0.0492
0.0448
0.0050
85.0
GraphBFN
0.0005
0.0294
0.0002
0.0046
0.0005
0.0560
0.0370
0.0053
87.5
DualDiff
0.0003
0.0275
0.0002
0.0038
0.0004
0.0473
0.0365
0.0042
85.0
Experiments ZINC250K dataset. Following previous studies, report the FCD and KI scores, where higher values indicate better performance. The results are reported from EDM- SyCo and the original papers. Methods that do not report the KL metric denoted N.A.
Table 2:
Method
FCD score (t)
KL score (t)
Novelty (1)
Uniqueness (t)
Validity (1)
GraphAF
0.05 + 0.00
0.67 + 0.01
0.91 + 0.01
0.91 + 0.01
1.00 + 0,00
Autoreg.
MoL.eR
0.83 + 0.00
0.97 + 0.00
0.99 + 0.00
0.99 + 0.00
1.00 + 0.00
GruphAr
0,04 = 0.00
N.A.
1.00 + 0.00
0.99 + 0.00
0.88 + 0.00
MiCaM
0.63 =.02 0.02
0.94 = 0.00
0.98 + 0.00
0.98 0.00
1.00 + 0.00
MAGNet
0.76 0.00
0.95 + 0.00
0.99 + 0.00
0.99 + 0.00
1.00 + 0.00
GDSS
0.10
N.A.
1,00 0.00
1,00 0.00
0.97 + 0.01
One-shot
DiGress
0.91 + 0.00
0.99 0.00
0.85 0.01
GruM
0.64
N.A.
1.00 0.00
0.99 0.00
SwinGNN
0.67 1 0.00
N.A.
0.96 1 0.00
0.91 1 0.00
EDM-SyCo
0.85 0.01
0.96 + 0.00
1.00 0.00
1.00 0.00
0.88 0.01
DualDiff
0.91 0.02
0.98 + 0.01
1.00 0.00
1.00 0.00
0.92 + 0.02
5.2 GENERIC GRAPH GENERATION
From Table T we can observe that our proposed DualDiff achieves competitive performance across diverse metrics. Essentially, generic graphs demonstrate pronounced geometrie properties. The topologically-enhane
global information can effeetively capture the core features of the dataset, thereby providing valuable guidance for the generation of local information. We further validate model's performance on the mainstream Community-smal Ego-small, and Grid datasets, as shown in Table 8 further illustrating the effectiveness and applicability of our model.
5.3
MOLECULAR GENERATION
We also evaluate DualDiff on molecular generation tasks. As shown in Table DualDiff surpasses other advanced methods on the FCD and KL metrics. and exhibits competitive performance on the novelty and validity metrics. Fundamentally, molecular structures demonstrate dependencies across multiple scales, encompassing both atomic dependencies within motifs and interactions between distinct motifs. By equipping the generative model with both global and local awareness, DualDiff can substantially improve the quality of the generated molecules. Besides, most methods achieve novelty and uniqueness scores approaching 100%, which can be primarily attributed the large molecular sizes. The resulting exponential growth in the number of possible molecular structures makes it difficult to sample molecules identical to those in the training and test sets.
Due to the page limits, we present the performance our model on the QM9 dataset in Appendix C.2 and the large-scale molecular dataset MOSES in Appendix C.4 In both cases, DualDiff
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achieves competitive results compared to other advanced methods, further demonstrating the effec- tiveness of our approach. addition to unconditional graph generation, Dual can also be applied to conditional generation tasks. Detailed experimental results provided in Appendix C.3. where DualDiff likewise demonstrates strong conditional generation performance.
5.4 COMPREHENSIVE COMPARISON WITH HIERARCHICAL MODELS
Hierarchical graph generation methods Karami (2024) ; Bergmeister et al. (2024), which model com- plex graphs by decomposing them into multiple levels and progressively generating structures from coarse to fine, have achieved notable success graph generation. contrast to these autoregressive and coarse-to-fine modeling approaches, our dual conditioning mechanism enables dynamic inter- action between global and local information. This allows for a more accurate capture of the joint distribution of global and local features compared to hierarchical methods. We provide compari- son of model performance between our method and hierarchical approaches, as shown Table further demonstrating the effectiveness of our approach.
Table 3: Comparison between advanced hierarchical models.
Methods Metries
Planar
SBM
Dog.
Clus.
Orbit
Spec.
Deg.
Clus.
Orbit
Avg.
[Bergmeister al, 2024
0.0003
0.0245
0,0006
0.0104
0.0119
0.0517
0.0669
0.0067
HiGen KKaram 2024
0,0012
0,0435
0.0234
0.0025
0.0017
0.0503
0.0604
0.0068
DualDit (Ours)
0.0003
0.0275
0.0002
0.0038
0.0004
0.0473
0.0365
0.0042
5.5 EXPERIMENTS ON GENERATING 3D MOLECULES
In this subsection, we will evaluate the effectiveness of our approach in 3D molecular tasks. Fol- lowing GEOLDM (Xu al. 2023), we replace the lightweight MLP decoder with an EGNN to ensure that both latent embedding and coordinates are simultaneously generated. The experimental results on the QM9 dataset are shown Table 4 Our method achieves competitive performance compared to the advanced GEOLDM and EQUIFM (Song al. 2023), which further illustrates the effectiveness of our methods on generating 3D molecules.
Table 4: Experiments 3D molecular generation on the QM9 Dataset.
Methods
Atom Sta (%)
Mol Sta (%)
Valid & Unique (%)
Data GEOLDM
99.0
95.2
97.7
98.9
89.4
92.7
EQUIFM
98.9
88.3
93.5
DualDiff
98.9
88.7
99.3
5.6 MODEL ANALYSIS
Ablation study. We conduct an ablation study the proposed dual conditioning mechanism, as shown in Table 5 Specifically, we compare the model performance of DualDiff without any conditioning, with self-conditioning, with only global-to-local conditioning. and with full dual con- ditioning on the ZINC250K dataset.
Table 5: Ablation study of dual conditioning.
The results demonstrate that dual ditioning can substantially improve the quality of the generated graphs and sig- nificantly outperforms the conventional self-conditioning approach, underscoring the importance of the interaction between global and local topological information.
Method
FCD
KL (?)
DualDiff (wlo any cond.)
0.65 + 0.01
0.82 0.02
DualDiff (w self cond.)
0.72 4 0.01
0.89 0.89 0.02
Difl (Z, + Z,)
0.75 + 0.03
0.95 0.95 0.02
DualDiff (2, 21)
0.83 + 0.01
0.95 + 0.01
DualDiff (Z, + Zi)
0.91 + 0.02
0.98 1: 0.01
Parameter Analysis. We conduct an extensive evaluation of model performance under varying key parameters, including the frequency p of process (i) during training, steps used in Section 4.3 and the number of clusters K. Detailed

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Published as a conference paper at ICLR 2026
results the ZINC250K dataset are presented in Figure B The left panel indicates that selecting a moderate value for facilitates effective interaction between global and local information. The middle panel shows that increasing during sampling enables the model focus local details, leading to enhanced performance. Finally, the right panel highlights the significance of appropriate cluster number K: smaller K not only reduces computational overhead but also preserves essential global information. Besides, we also include the detailed ablation studies of different clustering methods, and cluster numbers in Appendix C
complex
Model performance on p
Model performance on m
Model performance on K
1.0
PCD
Metrics
0.9
0.8
0.7
0.6
p
0.2
p
p=1
K=8 K=
Figure 3: Experiments of model parameters on ZINC250K dataset.
Model Efficiency. Apart from the advanced generation performance, model also demonstrates high efficiency. As shown in Table 6 DualDiff achieves competitive results using only approxi- mately 200 steps. We further evaluate the actual time to generate 10,000 molecular graphs the QM9 and ZINC250K datasets, as reported in Table where "w/o dual." refers to the time without dual conditioning. The results show that the dual conditioning mechanism introduces acceptable computational overhead, and our model consistently maintains high efficiency. Additionally, we provide further explanations and experiments regarding the efficiency of DualDiff Appendix C.7
Table 6: Model performance at different diffusion
Table Experimental evaluation of the model's sampling efficiency.
steps on Community-smal and Ego-small datasets.
Dataset
Models
50
100
200
500
1000
Dataset
Model
Sampling (s)
GDSS
0.110
0.104
1.172
0.061
0.046
GDSS DualDiff
109
Comm.
GSDM
0.109
0.079
0.021
0.011
0.009
QM9
DualDiff
0.098
0.043
0.009
0.007
0.006
w/o dual.
32
GDSS
0.031
0.023
0.021
0.017
GDSS
1870
Ego.
GSDM
0.044
0.026
0.024
0.019
0.016
ZINC250K
DualDiff
223
DualDiff
0.032
0.012
0.005
0.005
0.004
wwo dual.
LIMITATIONS AND FUTURE DIRECTIONS
The proposed DualDiff currently primarily designed for generating generic and simple molecular graphs. However, general paradigm, DualDiff can effectively model local and global informa- tion simultancously, and thus can be broadly transferred to other data with hierarchical characteris- tics, such proteins, peptides, and even circuit design. We consider these extensions as promising directions for future work and will explore the applicability of Difl to diverse domains.
7 CONCLUSION
In this paper, we design unified global and local topology-aware generative model, i.c., DualD- iff. DualDiff employs two-branch diffusion framework and dual conditioning mechanism, which can effectively model the joint distribution between global and local information, thereby naturally capturing the complex and entangled multi-scale relationships inherent in graphs. Empirical results demonstrate that our proposed model significantly enhances the quality of generated graphs while further ensuring training stability and sampling efficiency.
ACKNOWLEDGMENTS
The research work described in this paper was conducted the JC STEM Lab of Machine Learning and Symbolic Reasoning, funded by The Hong Kong Jockey Club Charities Trust.
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A MODEL DETAILS
A.1 AUTOENCODER AND LATENT SPACE
Encoder. In this subsection, present the details of the encoder. For generic graphs, we employ peessng graph neural networks (MPNNs) (Gilmer et al. 2017) to implement Eg. The latent representation Z for each node is computed via layer-wise message passing as follows:
_ UPDATE (8)
! (GG (Z : ( E(()))
(12)
where denotes the k-th layer, UPDATE and AGG correspond to the update and aggregation functions, and (e)) denotes the set of neighbors of node v. For molecular graphs, atomic coor- dinates provide informative representations of inter-node relationships. Therefore, we incorporate the equivariant graph neural network (EGNN) to implement Eg, enabling the mapping 9 Z and producing coordinate-enhancee node embeddings, i.e., Z (Z (()).
Decoder. To reconstruct graph-structured data from the latent embedding Zi and Zy. the decoder first combines both local and global information. Then, leverages lightweight MLPs separately reconstruct the graph features H and the adjacency matrix A. Specifically, the decoder first utilizes FiLM layer to integrate global and local information, obtaining global eenexxtennhnnee node embeddings, as follows:
7,8- = Linear(S, S Zy), Zi 7 = Z B,
(13)
where Sg E N is the affiliation matrix, which can be obtained based on the similarity between Zy and Z₁ Then, our model employs two separate lightweight MLPs recover graph-structured data, i.e., the node features and adjacency matrix. Formally, we have:
X MLP mode Zi).
(14)
À = (MLP ad) ZF))
(15)
where and B denote the scale and shift parameters induced by the global cluster embedding, is Sigmoid activation function, X and Á represent the reconstructed node and adjacency matrix. For molecular graphs, we can further leverage the coordinate information to facilitate more accurate adjacency matrix reconstruction, which can be formulated as follows:
A.j- = MT.Padt adj (Concat
(h) z
200h 1 11)
(16)
Latent Space. Unlike discrete graph generation paradigms, we follow the LGD paradigm by map- ping the graph space into a continuous latent space. This approach allows us to fully leverage in- formative node features to enhance the quality of the generated graphs. Besides, traditional discrete graph diffusion frameworks require explicit diffusion over the entire adjacency matrix, resulting in O(N2) complexity. In contrast, node embeddings in the latent space can implicitly model the re- lationships among nodes. Our latent diffusion approach operates at the node level, with diffusion space complexity of ONN), which significantly enhances computational efficiency.
A.2 GLOBAL INFORMATION EXTRACTION
We leverage powerful graph clustering methods to extract topology-aware global information. Specifically, for generic graphs without explicit coordinate information, we first perform spectral decomposition on the graph Laplacian matrix, L D A. yielding L = UAUT, We then select the leading eigenvectors of U and construct U = [u1. U2...., uk] E N The K-means algorithm is subsequently applied to the rows of U to obtain the final clustering results, denoted as Sy. Additionally, we incorporate the eigenvectors U into the node embeddings to inject meaningful topological information, which can largely enhance the reconstruction ability of the autoencoder.
For molecular graphs, we can utilize atomic coordinates to perform clustering by directly applying K-means to the alom-level coordinate representations, (=), The resulting cluster centers can be re- garded the centers of molecular fragments. This global structural information can provide mean- ingful guidance, steering molecular generation towards more interpretable and chemically relevant structures. After clustering and obtaining the affiliation matrix Sz, we aggregate the embeddings of nodes belonging to the same cluster and ultimately obtain Zg Kxd
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A.3 DENOISING NETWORK
In this subsection, we include a detailed description of the architectures used in the denoising net- works De, and Dø,- For general graph generation tasks, both Dø, and De, are implemented using the Graph Transformer, which enables global all-pair message passing by aggregating information from all node embeddings to update each node's representation. For molecular generation tasks, we adopt EGNN for De, and Da, to ensure the equivariance of the generation process.
A.4 DUAL CONDITIONING MECHANISM
In this subsection, we include more details of the proposed dual conditioning mechanism. During the two processes of the dual conditioning mechanism, we can leverage the node-to-cluster similarity achieve dynamic global and local information exchange without explicit clustering. Specifically, we first obtain the estimated outputs from the previous step: Z,, Do, (Zi, Ci and Zg,0 Da, gZ,, Cg). This step is analogous to self-conditioning. where the output from the previous step is used condition for the current step. Then
calculate the affiliation matrix Sn E N based on 2д,0 and Zg.0. where Si 1, if j arg max, sim 2i Z2,0) and S O, otherwise. During process (i), assign each node to its corresponding cluster according Sg. and introduce the condition based on (9). During process (ii) in (TO), we utilize message passing and average
pooling operations to extract the initial global condition C, where C ΣM= 24
Σi- 5
B EXPERIMENTAL SETTING
B. METRICS
In Table we report the scores of the FCD and KL metrics following (Brown et al. 2019). Specifi- cally, the detailed calculation methods are as follows:
KL score. For both generated and reference molecules, we calculate the following descriptors: BertZT, MolLogg,, MolWt, TPSA, NumHAcceptors, NumHDonors, NumRotatableBonds, Nu- mAliphatticRinss NumAromaticRings. and the ECFP4 fingerprint-based similarity to the nearest neighbor. The KL divergence DKL is computed for each descriptor to measure the difference be- tween the two molecular sets. These divergences are then combined to produce final normalized score, given by 1/ E exp(-DKL.)).
FCD score. FCD is evaluated using the hidden representations from ChemNet, a neural network trained to predict biological activities. Specifically, we extract the means and covariances of the last hidden activations in ChemNet for both the generated and reference molecules. The Fréchet distance between these two distributions is then calculated. To obtain a final score between 0 and 1, this distance is normalized using exp( exp(-0.2 -CDD
B.2 DATASETS
Following previous methods, leverage eight graph datasets to evaluate our proposed DualDiff extensively. The datasets include: (1) Ego-small: 200 small ego graphs collected from Citeseer graph datasets (Sen et al.)
2008); (2) Community-small: 100 random graphs generated by (ERDdS)
& R&wi 1959; (3) Grid: 100 standard 2D grid graphs; (4) Planar: 200 synthetic planar graphs
with N 64 nodes; (5) SBM: 200 stochastic block model graphs with 2-5 communities, where
44 < N < 187 and each community contains 20-40 nodes; (6) ZINC250K: 249,455 molecular graphs with 9 node types (Irwin et al. 20122;; (7) QM9: 133,885 molecular graphs with 4 node types Ramakrishnan et al. 2014); (8) MOSES: lcage-scale molecular dataset (Polykovskiy et al. 2020).
B.3 TRAINING DETAILS
All experiments conducted NVIDIA 4090 GPUs. We employ the Adam optimizer with learning rate selected from {1 x 10 3, x 10 10-4, 1 x 10 40- The sampling steps are fixed at 256. Our parameters follow the settings in the Elucidated Diffusion Models (EDM), with Zwin 0.002,
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maa 80, and p = 7. The hidden dimensions of Zi and Zg set to d {32, 64, 128). Besides, choose 2nd order Heun the ODE solver. To ensure equivalence in molecular generation tasks, we also subtract the center of mass from the noise added to the coordinates at cach step. Each experiment is repeated three times, and report the average results.
For molecular generation, the encoder utilizes a 2-layer EGNN architecture, while a 2-layer MPNN used for generic graph generation. The decoder is composed of two separate MLPs, responsible for reconstructing edge and node features, respectively. For molecular tasks, Do, and Do, are both implemented by 4-layer EGNNs; for generic graphs, we employ 4-layer Graph Transformers.
C ADDITIONAL EXPERIMENTAL RESULTS
C.I EXPERIMENTS ON OTHER GENERIC GRAPH DATASETS
In this subsection, we further evaluate the model's performance the Ego-small, Community- small, and Grid datasets, as shown in Table g From Table R] it can be clearly observed that our proposed method achieves competitive performance across four widely used generic graph gener- ation datasets. For Ego-small and Community-smm patterns, our proposed DualDiff can effectively capture both global and local information, exhibiting advanced generation performance. Besides, on larger datasets such as Grid datasets, DualDiff main- tains strong performance, further demonstrating its scalability and robustness. Essentially, generic graphs typically exhibit prominent global information. Incorporating global awareness during the generation process can effectively guide the generation process toward a meaningful direction, ac- celerate training convergence, and improve the final quality of the generated results. To further demonstrate the effectiveness of our method, we also include visualization results of graphs gener- ated by DualDiff in Appendix
datasets that display clear global topological
Table 8: Experiments generic graph generation. We report MMD distances of different statistics between the test set and the generated graphs. Avg. means the average results of three metrics.
Ego-small
Community-small
Grid
Methods
Real, S [V]
Synthetic, 12
[V] S
Symthetic, 100
[V| S
Deg.
Clus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
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 (hh al.
0.034
0.110
0.001
0.047
0.181
0.205
GraphDF 1.00 or ar. 2021
0.041
0.135
0,010
0.060
0.064
0.125
0.037
0.070
GDSS
0.021
0.024
0.007
0.017
0.045
0.086
0.007
0.046
0.111
0.005
0.070
0.062
DiGress
0.025
0.021
0.008
0.018
0.051
0.083
0.009
0.048
0.189
0.003
0.075
0.089
GSDM
0.021
0.022
0.007
0.016
0.011
0.015
0.001
0.009
0.002
0.0
0.0
0.0007
GruphBFN
0.011
0.014
0.002
0.009
0.002
0.060
0.045
0.011
0.040
0.032
DualDiff (Ours)
0.007
0.008
0.001
0.005
0.007
0.012
0.002
0.007
0.001
0.0
0.0
0.0003
C.2 EXPERIMENTS ON QM9 DATASET
To comprehensively evaluate the effectiveness of our model on molecular generation tasks, we also followed the experimental settings of GDSS. In addition the FCD metric, we evaluated the neigh- borhood subgraph pairwise distance kernel (NSPDK) MMD (Costa & De Grave 2010), which mea- sures the graph kernel distance between subgraph structures and node features. The results, as shown in Table 9 demonstrate that our proposed DualDiff model outperforms other state-of-the-a methods, further highlighting its competitive performance in molecular generation tasks.
C.3 EXPERIMENTS ON PROTEINS DATASET
To further validate the effectiveness of our methods on large generic graphs, compare DualDiff with other advanced generative methods, as shown in Table I0 The results further illustrate the effectiveness and applicability of our methods.
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Table 9: Experiments of different generative methods on QM9 dataset.
Method
FCD (})
NSPKD (1)
Unique (7)
Valid w/o cor. (1)
GDSS
2.900
0.003
GSDM
2.650
0.003
0.97
0.99
Graphusion
2.593
0.004
0.99
0.97
DiGress
0.095
0,0003
GruM
0.108
0.0002
0.97
0.99
LGD LGD Zhou et al. 2024
0.100
0.0002
0.98
0.98
0.093
0,0002
GraphBBN
0.101
0.0002
0.98
0.99
DualDiff (Ours)
0.092
0.0001
0.97
0.98
Table 10: Experiments of diffrent methods on the Proteins Dataset.
Methods
Deg.
Clust.
Orbit
Spec.
GraphRNN
0,004
0.1475
0.5851
0.0152
DiGress
0.0041
0.0489
0.1286
0.0018
GDSS
0.0861
0.5111
0.732
0.0748
GruM
0.0019
0,0660
0.0345
0.0030
DualDiff
0.0022
0.0309
0.0067
0.0030
C.4 EXPERIMENTS LARGE-SCALE GENERATION
To further illustrate the effectiveness of our methods in large-scale generation scenarios, we also evaluate our methods on the MOSES datasets (Polykovskiy et al. 2020). From Table [T our pro- posed DualDiff achieves comparable performance with advanced DiGress and LGD in terms of validity and novelty metries. Besides, DualDiff can further bridge the gap between diffusion-based graph generation models and other traditional methods, illustrating its effectiveness.
Table 11: Experiments of large-scale generation on MOSES dataset.
Model
Class
Valid
Unique
Novelty
FCD (L)
JT-VAF Jin
2018
Fragment
100
100
99.9
1.00
GraphINVENT Mcrcado
2021
Autoreg.
96.4
99.8
1.22
ConGress Vignac el
One-shot
83.4
99.9
96.4
1.48
DiGress Vignac al.
One-shot
85.7
100
95.0
1.19
LGD Zhou et at. 2024
One-shot
97.4
100
95.9
1.42
GraphBFN (Sngg al. 2025b
One-shot
88.5
99.8
89.0
1.07
DualDiff (Ours)
Onc-shot
96.8
100
96.2
1.16
C.5 EXPERIMENTS ON CONDITIONAL GENERATION
We also extend proposed DualDiff to conditional generation scenarios. Following the settings in GEOLDM and LGD, we evaluate the controllable molecule generation given the desired property value We domain-specific condition encoder obtain the embedding of the property conditions Ts)). Then, the autoencoder and denoising networks take the condition embeddings as additional inputs, shifting the latent representations towards the data distribution aligned with the condition. Specifically, following GEOLDM, we split QM9 dataset into two parts, each including 50k molecules. Then, we train a property prediction network and our diffusion model on these two halves, respectively. For evaluation, we generate samples given the property and leverage the property prediction network to predict the property the generated samples. Finally, we report the Mean Absolute Error (MAE) between and to measure the controllable performance of DualDiff. In this experiment, we consider six important chemical properties: polarizability a, orbital energies EHOMO, ELUMO and their gap As, dipole moment μl, and heat capacity C., From Table 12 DualDiff demonstrates competitive controllable generation performance with advanced LGD and GEOLDM, further illustrating its effectiveness conditional generation tasks.
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Table Conditional generation results on QM9 (MAF. +
Model
CHOMO
(LUM
A
Co
Random
1.616
9.01
645
1457
1470
6.857
GeoLDM
1.108
2.37
340
522
587
1.025
LGD
0.879
2.43
313
641
586
1.002
DualDiff (Ours)
0.834
2.35
322
587
582
0.987
C.6 ABLATION STUDY ON DIFFERENT CLUSTERING METHODS
To further validate the effectiveness of our model with different clustering algorithms, we evaluate the impact of various clustering methods the QM9 dataset, shown in Table [3] The results demonstrate that our approach remains effective across different clustering strategies.
Additionally, we discuss the influence of the number of clusters model performance. We find that adopting a moderate number of clusters achieves a good balance between efficiency and accuracy, shown in Table [4. which underscores the importance of selecting an appropriate cluster number.
Table 13: Experiments of different clustering methods.
Clustering Method
FCD (1)
NSPKD (1)
Unique (f)
Valid w/o cor (f)
Spectral Clustering
0.094
0.0004
0.97
0.98
GMM
0.095
0.0006
0.96
0.97
Louvain Algorithu
0.100
0.0003
0.96
0.97
K-means (Current)
0.092
0.0001
0.95
Table 14: Experiments on the number of clusters.
Methods Metries
Enzymes
Planar
Deg.
Clus.
Orbit
Avg.
Deg.
Clus.
Orbit
Avg.
4
0.015
0.109
0.009
0.0443
0.0008
0.0415
0.0009
0.0104
6
0.013
0.111
0.010
0.0447
0.0003
0.0275
0.0002
0.0038
8
0.010
0.083
0.007
0.0333
0.0004
0.0271
0.0003
0.0064
10
0.012
0.087
0.007
0.0353
0.0003
0.0291
0.0002
0.0056
C.7 FURTHER EXPLANATION OF DUAL CONDITIONING MECHANISM
To further illustrate the effectiveness of the dual conditioning mechanism, we compare the model performance with and without the dual conditioning mechanism under varying diffusion steps. The detailed results are shown Table [5] We can observe that introducing the dual conditioning mech- anism can achieve strong performance even with very small number of steps, further highlighting the importance of information exchange between global and local representations.
C.8 EXPLANATION OF THE MODEL EFFICIENCY
To further validate the overhead of the additional dual conditioning introduced in DualDiff, we conduct analyses from two perspectives: complexity analysis and empirical evaluation of practical sampling time and memory consumption.
D EXPLANATION OF THE CLUSTERING METHODS
In the global information extraction, we perform clustering in either the 3D geometric space or the topological space of the graph to obtain topologically-base classes. By combining these geometri- cally meaningful clusters with node embeddings, we introduce a topological bias, which facilitates the extraction of effective global information. In general, the geometry topology of the graph serves as a powerful, informative, and critical indicator, further ensuring the effectiveness of the global information extraction.
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Table 15: Model performance of different conditioning methods at distinct diffusion steps.
Dataset
Models
50
100
200
500
1000
DualDiff (w/o any cond.)
0.099
0.078
0.032
0.025
0.024
Comm.
DualDiff (w. self' cond.)
0.100
0.064
0.018
0.016
0.016
DualDiff
0.098
0.009
0.006
To further validate the effectiveness of the clustering methods used, compare them with ad- vanced learnable deep clustering methods. Learnable deep graph clustering or pooling methods ex- tract global information based on the similarity of node embeddings, typically leveraging K-means clustering or pooling to obtain global representations. The key difference between our approach and these methods is that extract global information in training- free manner by utilizing geometry- based clusters, whereas deep clustering methods depend on the similarity of latent embeddings and require additional training time and computational resources. We provide comparison between deep clustering and our approach, as shown below. Notably, K-means and spectral clustering achieve performance comparable to deep clustering, further demonstrating the effectiveness of topological bias in capturing global information.
E EXPLANATION OF THE SAMPLING PROCESS
E. THEORETICAL INSIGHTS OF THE DUAL CONDITIONING
To model the high-dimensional and intractable joint probability distribution P(Zi, Zg). diffusion- based methods often use the target Z,o and Zg.0 as conditions to model the conditional probability path pt. This can be factorized into the following three equivalent forms:
pr (2, 2g)
(e Zg Zi,0 2g00dZZ,,dZ2.00
(17)
Zi,0. odaat
p7 P.(ZiiZ,,))p 2,
(18)
Pr(2i Zoo), (2.
(19)
Zi,0. Z1,0,Z,,orda
These three factorizations provide alternative ways to model the joint conditional probability path p: 1) simultaneously incorporating both global and local conditions; 2) incorporating local information into the generation of global information; incorporating global information into the generation of local information.
Furthermore, in diffusion models, P: is typically instantiated example, in the decomposition (r Z1,0), Pe( T Zz Zg, Pr( can be modeled as NZZ pe (Z),))) (Zt,))) corresponding to a standard diffusion process without additional condi- tions, i.c., C₁ = 0 in process (ii). Besides, Pz Zi, Zg.o) = NZzg Pe Z1, Zg.o), oz Zg.o)), which can be seen as conditional diffusion process with Zt as the condition. By further in- tegrating self-conditioning we obtain C (Z,,, Zg,0). Similarly, process (i), (Ci, C2) = (((2),0. 2,,0), 0)) corresponds to modeling pr (Z) Zg (Z,1
Gaussian path. For
E.2 EFFECTIVENESS OF THE ALTERNATING MECHANISM
During the sampling phase, we take inspiration from the server-client communication paradigm and employ a strategy that alternates m steps of process (i) with a single step of process (ii). In this section, we provide a more detailed explanation of this approach. As illustrated in Figure process (i) incorporates global information into the generation of local details, analogous to information flow from the server to the client for local updates. Conversely, during process (ii), local informa- tion is aggregated to improve the prediction of the global representation, which can regarded communication from the client the server for global aggregation. To ensure
stable
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sampling process, perform local updates for each global aggregation, which enhances overall performance and facilitates the sampling procedure.
In addition, this asymmetric alternating design is primarily intended to ensure stability during the sampling process and to promote a better match between global and local information. Essentially, the learning difficulty of global and local information is different. Generally speaking, learning local details is more complex than capturing global information. The asymmetric paradigm allows the sampling process focus more integrating global information into local details. In contrast, a symmetric design would introduce both global and local information into the learning process simultaneously during sampling. However, the quality of the generated local information may be significantly lower than that of global information, which may provide incorrect guidance to the global process and cause instability. To further validate the effectiveness of this asymmetric design, we conducted ablation studies on the asymmetry design, as shown in Table T6. which demonstrate the rationality of the asymmetric alternating strategy.
Local update
Global aggregation
Z client
Zg
process (I)
process (ii)
Figure Explanation of the sampling process.
Table 16: Comparison asymmetric and symmetric methods
Datasets
Enzymes
Planar
asymmetric
0.010
0.083
0.007
0.0003
0.0275
0.0002
symmetric
0.018
0.107
0.009
0.0009
0.0425
0.0004
With respect to convergence, the global information provides a "big picture" for the generation of local details, facilitating convergence and constraining the exploration space for local details. We further compare the performance of simultaneous and alternating methods under the same number of sampling steps. The results show that our approach can achieve superior performance with much smaller number of steps.
F
ETHIC STATEMENT
This work focuses on the development of novel algorithms for graph generation, with the primary goal of advancing research in machine learning. Our methods intended to facilitate the design and discovery of new molecular structures and graph-based data representations, which may be beneficial for applications materials science, drug discovery, and related fields. We believe that our work does not introduce negative social or ethical impacts.
G REPRODUCIBILITY STATEMENT
In this paper, we provide detailed model configurations, as presented in Appendix A Besides, we also provide comprehensive experimental details, as shown in Appendix Additionally, the datasets used the paper are all publicly available, ensuring consistent and reproducible evaluation results. The code will be available at https://ithuuh.cm//////iuuuu
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H
LLM USAGE
Large language models (LLMs) were utilized to support the writing and editing process of this manuscript. particular, we employed an LLM help refine the language. enhance readability, and ensure clarity throughout various sections of the paper. It is important to note that the LLM was involved in the ideation, research methodology, or experimental design. The authors take full responsibility for the content of the manuscript, including any text generated or polished by the LLM. We have ensured that the LLM-generated text adheres to ethical guidelines and does not contribute to plagiarism or scientific misconduct.
I VISUALIZATION
To provide a more intuitive demonstration of the effectiveness of our method, we present visualiza- tion results of DualDiff on datasets including Ego-small, Community-small, QM9, and ZINC250K. As shown in Figure 5 Figure Figure and fully capture the distinctive characteristics of each dataset, further highlighting the strong effective- ness and practical utility of our approach.
and Figure the graphs generated by method success-
e=1, n=5
e=3, n=4
e=4, n=d
e=3
e=5, n=6
n-6
0-3,
em3, nmd
e=18, nm12
em5, nm6
em3 nm4
em6, nm6
n=10
e=7,
e=1,n=7
Figure 5; Visualization results on Ego-small dataset.
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6-40, n-18
6-49, n-18
e-49, n-18
0-21, n=12
e-50, n=18
2m11, n=12
2=27, n=14
50 n=18
e= n=14
e= n=
C-38, n-16
c-37, n-16
e-21, n-12
e-50, n-18
c-21, n-12
Figure 6: Visualization results on Community-sman small dataset.
HO
NH
HO
NH
Figure 7: Visualization results on QM9 dataset.
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warde
Figure 8: Visualization results ZINC250K dataset.
25


Reviews:
[{"review_id": "w9WRkU31FX", "reviewer": "Reviewer_o5es", "summary": "This paper proposes DualDiff, a dual-branch latent diffusion model that learns both global and local graph structures. The key idea is to run two diffusion processes, one at the node level and one at the cluster level, and let them interact through a dual conditioning mechanism. This design helps the model capture both fine-grained details and overall topology. Experiments on multiple benchmarks show clear gains over prior graph diffusion models, proving that the method effectively models multi-scale dependencies in graphs.", "questions": "1. The paper mentions that the alternating scheme improves stability, but is there any quantitative or theoretical evidence to support this? What happens if the two processes are trained simultaneously, rather than alternately?\n2. How sensitive is the model to the choice of clustering algorithm or the number of clusters K? Did you try other global extraction methods besides K-means or spectral clustering?\n3. Computation and scalability: Since you have two denoising networks and alternating updates, how much additional compute or memory does this introduce compared to a standard latent diffusion model, such as Latent Graph Diffusion?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Good motivation and clear design idea\nThe paper starts from a real problem that most graph diffusion models only handle node-level stuff. Splitting the process into local and global branches makes sense and is well explained.\n2. Thoughtful architectural design\nAlternating between the two diffusion branches is clever. It’s like doing local updates and then global aggregation, which keeps the training stable and avoids conflicts between the two processes.\n3. Strong and broad experiments\nThey test on both synthetic graphs and real molecular datasets, and the gains are consistent. It shows the method isn’t overfitted to one type of data.", "weaknesses": "1. too dependent on clustering\nThe global branch comes from clustering nodes, so if the clustering is poor, the “global info” might be misleading. They don’t analyze this sensitivity much.\n2 . Efficiency claim not fully convincing\nThey say the model is efficient in the latent space, but with two diffusion networks and alternating steps, it’s unclear how much heavier it actually is.\n3. No solid theoretical grounding for stability\nThe alternating process is explained intuitively (like server-client updates), but there’s no formal guarantee or analysis of convergence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "01L41rf1kv", "reviewer": "Reviewer_rfZi", "summary": "This paper proposes a latent graph diffusion model that concurrently denoises node-level and cluster-level representations, coupled with a bidirectional conditioning mechanism to exchange global and local topological cues during generation. Extensive experiments on eight generic and molecular graph benchmarks demonstrate competitive or state-of-the-art performance in both unconditional and property-conditional generation while requiring fewer diffusion steps than prior diffusion counterparts.", "questions": "N/A", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n2. Using global & local latent features in graph diffusion models with learnable cross-conditioning is a reasonable design for the task.\n3. The empirical results across diverse datasets show the effectiveness of the proposed method.", "weaknesses": "1. The paper uses different global feature extraction methods in different tasks, but how to choose the method is unclear, which may lead to difficulty in generalization to new tasks.\n2. In ablation study, what if there is only local-to-global condition?\n3. At the beginning of sampling process, the number of nodes and clusters should be pre-determined, how to decide these numbers? What about the generalization ability of these numbers?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "XusEKTghDS", "reviewer": "Reviewer_Cc24", "summary": "This paper proposes DualDiff, a latent diffusion model designed for graph generation that jointly captures both local and global topological information through a two-branch diffusion process and a dual conditioning mechanism.\n\nThe model encodes input graphs via a pretrained graph autoencoder, extracts global representations via graph clustering (spectral or K-means), and performs parallel diffusion on node-level and cluster-level latent variables, alternately conditioning one on the other. Experiments are conducted on generic graph datasets (Ego-small, SBM, etc.) and molecular datasets (ZINC250k, QM9, MOSES).", "questions": "1. Why do the authors believe that a fixed, non-learnable clustering method (K-means or spectral) can define “global topology” more effectively than end-to-end hierarchical pooling approaches in modern GNNs?\n\n2. How sensitive are the results to the choice of clustering algorithm and the number of clusters $K$?\n\n3. Why not train the clustering jointly with the autoencoder or in the diffusion process to obtain adaptive global representations?\n\n4. The model’s validity on ZINC250k is only 92%. Does this reflect an inherent inability to capture local chemical rules? Have the authors examined failure cases?\n\n5. Can the framework be extended to explicitly generate 3D molecular structures rather than only 2D topologies?\n\n6. Is there any theoretical basis to justify Eq. (8) as a valid probabilistic factorization of $p(Z_l, Z_g)$?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "- The motivation—capturing multi-scale (global and local) structural information in graph generation—is both timely and important.\n\n- The paper is very well-written and full of details.\n\n- The authors conduct comprehensive experiments, comparing with many recent diffusion-based and autoregressive graph generators.", "weaknesses": "1. The paper defines “global information” simply as the cluster centroids obtained from node embeddings via K-means or spectral clustering. This is a very coarse, heuristic, and outdated approximation of global structure. It is not truly “topology-aware” as claimed. The authors do not justify why a fixed, non-learned clustering is preferable.\n\n2. The statement that “existing methods still leverage node-level generative paradigms” is too absolute. Many recent approaches already incorporate hierarchical or subgraph-level generation.\n\n3. Although the authors emphasize that the encoder takes 3D coordinate information as input for molecular graphs, the model ultimately generates only 2D molecular structures (bond graphs) rather than full 3D geometries.\n\n4. The authors claim Eq. (7) “implies that modeling the joint distribution of global and local information can be decomposed as two complementary processes” . In fact, Eq. (7) is merely a standard conditional-probability identity—any joint distribution can be written that way. Therefore, it provides no theoretical evidence that the proposed alternation scheme meaningfully models $p(Z_l​,Z_g​)$.\n\n5. On ZINC250k (Table 2), the reported Validity = 92%, which is much lower than baselines such as GruM (99%), GraphArm (100%), and GDSS (97%).  This indicates a serious failure in generating chemically valid molecules and undermines the claim that DualDiff captures local chemical constraints (e.g., valence, functional groups).\n\n6. Protein datasets, which contain larger graphs (100 < |V| < 500) are also widely used in generic graph generation. Moreover, related baselines such as GruM have reported results on these datasets. The authors are encouraged to include experiments on protein graphs to more comprehensively demonstrate the effectiveness and scalability of their proposed approach.\n\n7. The paper does not provide a code release or supplementary implementation details.  \nGiven the model complexity and multi-stage training (autoencoder + dual diffusion), reproducibility is questionable. Code should be provided for verification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)