Global and Local Topology-Aware Graph Generation via Dual Conditioning Diffusion

Conference: ICLR 2026
OpenReview: https://openreview.net/forum?id=IZV9k5BGxi
Code: To be confirmed
Area: Graph Generation / Latent Diffusion Models
Keywords: Graph Generation, Latent Diffusion, Global-Local Dependencies, Bidirectional Conditioning, Molecule Generation

TL;DR

DualDiff decomposes graphs into node-level (local) and cluster-level (global) diffusion branches. By employing a "bidirectional conditioning" mechanism, global and local information alternately serve as conditions during the denoising process. This joint modeling of \(p(Z_l, Z_g)\) in a unified latent space significantly improves the generation quality of both general and molecular graphs.

Background & Motivation

• Background: Diffusion models have become the mainstream paradigm for graph generation. Latent space diffusion (e.g., GEOLDM, LGD, EDM-SyCo) further moves the process from the raw graph space to a low-dimensional latent space, balancing efficiency and scalability.
• Limitations of Prior Work: Most existing methods focus on "node-level" generation—treating nodes as independent entities during denoising, which struggles to capture dependencies within and between substructures. A few works incorporating global information (e.g., SubgDiff's subgraph prediction, Graphusion's cluster pseudo-labels) treat it only as one-way guidance, ignoring inter-substructure dependencies and failing to model the joint distribution of global and local features.
• Key Challenge: Graph data naturally exhibits multi-scale coupled dependencies. Molecules contain both local details (functional groups) and global topologies (spatial distribution of groups); these interact with each other. Single-way or single-scale modeling cannot adequately describe both levels simultaneously.
• Goal: Design a unified generative model capable of dynamically capturing both global and local topological information to model their joint distribution.
• Core Idea: "Joint Distribution Bidirectional Decomposition". Based on the identity \(p(Z_l, Z_g) = p(Z_l|Z_g)p(Z_g) = p(Z_g|Z_l)p(Z_l)\), the joint modeling is decomposed into two complementary processes: "global-to-local" and "local-to-global." A bidirectional conditioning mechanism allows the two diffusion branches to alternately condition on each other, achieving dual topology awareness.

Method

Overall Architecture

DualDiff is a two-stage latent diffusion framework. First, a pre-trained graph autoencoder maps the graph \(G=(H,A)\) to a unified latent space to obtain local representations \(Z_l \in \mathbb{R}^{N \times d}\), then clusters \(Z_l\) to aggregate global representations \(Z_g \in \mathbb{R}^{K \times d}\). Subsequently, a dual-branch diffusion (node-level + cluster-level) is performed in the latent space. A bidirectional conditioning mechanism enables information exchange during denoising. Finally, the decoder reconstructs the graph from the joint \(\hat Z_l, \hat Z_g\).

flowchart LR
    G[Input Graph/Molecule G] -->|EGNN/GIN Encoder| Z[Latent Representation Z]
    Z --> Zl[Local Repr. Z_l Node-level]
    Z -->|Clustering+Pooling GlobalExtraction| Zg[Global Repr. Z_g Cluster-level]
    Zl --> DL[Local Denoising Dθl]
    Zg --> DG[Global Denoising Dθg]
    DG -. Global→Local Condition .-> DL
    DL -. Local→Global Condition .-> DG
    DL --> Zlh[Denoised Ẑ_l]
    DG --> Zgh[Denoised Ẑ_g]
    Zlh --> Dec[Decoder Dψ]
    Zgh --> Dec
    Dec --> Gout[Generated Graph/Molecule]

Key Designs

1. Global Information Extraction: Explicitly encoding "substructure topology" into the global branch via clustering. The framework encodes the graph into latent space for node-level \(Z_l\), while \(Z_g\) is derived via clustering—the source of global topology awareness. Different strategies are used: Molecule graphs use K-means in atomic coordinate space for geometrically enhanced labels; general graphs use spectral clustering of Graph Laplacian eigenvectors for community partitioning. Given the assignment matrix \(S_g \in \{0,1\}^{N \times K}\), cluster-level embeddings are obtained via \(Z_g = \mathrm{Pooling}(S_g, Z) \in \mathbb{R}^{K \times d}\). Thus, \(Z_g\) carries long-range dependencies such as "which nodes belong to the same substructure" and "how substructures are distributed."

2. Dual-Branch Diffusion: SDEs for node-level and cluster-level branches. Given the different topological characteristics, DualDiff defines separate forward SDEs for \(Z_l\) and \(Z_g\) in the latent space: \(\mathrm{d}Z_{l,t}=f_{l,t}\mathrm{d}t+s_{l,t}\mathrm{d}W_{l,t}\) and \(\mathrm{d}Z_{g,t}=f_{g,t}\mathrm{d}t+s_{g,t}\mathrm{d}W_{g,t}\). Within the EDM framework, the drift term is set to 0 and the diffusion term \(s_{l,t}=s_{g,t}=\sqrt{2t}\). Two GNN denoising networks \(D_{\theta_l}, D_{\theta_g}\) are trained to recover clean latent representations: \(\mathbb{E}[\|D_{\theta_l}(\tilde Z_l,\sigma)-Z_{l,0}\|^2 + \|D_{\theta_g}(\tilde Z_g,\sigma)-Z_{g,0}\|^2]\).

3. Bidirectional Conditioning: Approximating the joint distribution via alternate conditioning. This is the core contribution. Based on the decomposition \(p(Z_l,Z_g)=p(Z_l|Z_g)p(Z_g)=p(Z_g|Z_l)p(Z_l)\), two complementary processes are defined: (i) Global-to-local (\(p(Z_l|Z_g)\)) and (ii) Local-to-global (\(p(Z_g|Z_l)\)). Practically, self-conditioning provides the previous step's predictions \(\hat Z_{l,0}, \hat Z_{g,0}\). The model switches between processes with probability \(p\): Process (i) uses \((C_l,C_g)=((\hat Z_{l,0},\hat Z_{g,0}),0)\), while process (ii) uses \((0,(\hat Z_{l,0},\hat Z_{g,0}))\). Process (i) uses FiLM-inspired modulation: \(\hat Z_{g,0}\) generates cluster-specific scaling/shifting parameters \(\gamma_i, \beta_i\) to modulate \(Z_l\) based on node-cluster similarity. Process (ii) compresses local details into a global condition using message passing and pooling: \(C=\mathrm{Linear}(\mathrm{Pool}(\mathrm{MP}(\hat Z_{l,0})))\).

4. Alternating Sampling: Stabilizing generation via a "server-client" approach from Federated Learning. Inspired by central servers aggregating client updates, global clusters are treated as servers and nodes as clients. Process (i)/(ii) correspond to local updates and global aggregation, respectively. During sampling, process (ii) is triggered only every \(m\) steps of process (i) (e.g., when t % (m+1) == 0). This scheduling significantly improves stability and quality.

Key Experimental Results

Main Results

General Graph Generation (Planar / SBM, lower MMD is better, higher V.U.N. is better):

Model	Planar Clus.↓	Planar Spec.↓	Planar V.U.N.↑	SBM Deg.↓	SBM Spec.↓
DiGress	0.0372	0.0106	75.0	0.0013	0.0400
GruM	0.0353	0.0062	90.0	0.0007	0.0050
GraphBFN	0.0294	0.0046	96.7	0.0005	0.0053
DualDiff	0.0275	0.0038	97.5	0.0004	0.0042

Molecule Generation (ZINC250k, higher FCD / KL is better):

Method	FCD↑	KL↑	Novelty↑	Validity↑
DiGress	0.65	0.91	0.99	0.85
GruM	0.64	N.A.	1.00	0.99
EDM-SyCo	0.85	0.96	1.00	0.88
DualDiff	0.91	0.98	1.00	0.92

3D Molecule Generation (QM9): DualDiff achieves 99.3% on Valid & Unique, outperforming GEOLDM (92.7%) and EQUIFM (93.5%).

Ablation Study

Ablation of Bidirectional Conditioning (ZINC250k):

Configuration	FCD↑	KL↑
No conditioning	0.65	0.82
Self-conditioning	0.72	0.89
Only \(Z_l \to Z_g\)	0.75	0.95
Only \(Z_g \to Z_l\)	0.83	0.95
Bidirectional \(Z_g \leftrightarrow Z_l\)	0.91	0.98

Key Findings

• Bidirectional exceeds self-conditioning: Improving FCD from 0.72 to 0.91 proves that global-local interaction is the primary source of quality gain.
• Directional value: \(Z_g \to Z_l\) (global guiding local) is more effective than \(Z_l \to Z_g\), though bidirectional is needed to approximate the joint distribution.
• Superiority over hierarchical methods: Compared to autoregressive/coarse-to-fine methods (HiGen, PPGN), DualDiff's dynamic interaction captures the joint distribution more effectively.
• Optimization of parameters: Moderate values for \(p\) and larger \(m\) (focusing on local details during sampling) yield better performance with a small cluster number \(K\).
• Efficiency: Competitive results are achieved in ~200 steps with manageable overhead from bidirectional conditioning.

Highlights & Insights

• Probabilistic grounding: The entire design is derived from the expansion of \(p(Z_l, Z_g)\), giving the bidirectional mechanism a clear probabilistic interpretation rather than being just an engineering heuristic.
• Reusing self-conditioning semantics: The "zeroing out" of one branch during training aligns with the robustness design of self-conditioning, allowing seamless integration.
• Federated Learning analogy: Mapping global clusters to "servers" and nodes to "clients" provides an intuitive rationale for the \(m:1\) sampling schedule.
• Generality: The framework is versatile, compatible with EGNN (for SE(3) equivariance in 3D molecules) or GIN/GCN (general graphs).

Limitations & Future Work

• Dependence on clustering: Global information relies on K-means or spectral clustering; poor clustering could negatively impact the global branch.
• Hyperparameter sensitivity: \(p, m, K\) require tuning for different datasets.
• Validity gap: While FCD/KL are high, Validity on ZINC250k (0.92) is still lower than GruM (0.99) or MoLeR (1.00).
• Two-stage training: The latent space is fixed after autoencoder pre-training; end-to-end optimization remains unexplored.

Related Work & Insights

• Latent Graph Diffusion (GEOLDM / EDM-SyCo): DualDiff extends the "autoencode-then-diffuse" paradigm from a single branch to dual branches.
• Global-informed generation (SubgDiff / Graphusion): These use one-way guidance; DualDiff addresses the lack of joint distribution modeling.
• Self-Conditioning: DualDiff upgrades single-scale self-conditioning to a cross-scale bidirectional mechanism.
• FiLM (Feature Modulation): Used as a tool to inject cluster-level global information into local nodes.

Rating

• Novelty: ⭐⭐⭐⭐ — Clear probabilistic derivation for dual-branch interaction.
• Experimental Thoroughness: ⭐⭐⭐⭐ — Comprehensive testing across 8 datasets including general and 2D/3D molecular graphs.
• Writing Quality: ⭐⭐⭐⭐ — Logical flow from motivation to sampling schedules.
• Value: ⭐⭐⭐⭐ — A generalizable multi-scale modeling paradigm for complex graph structures.

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