Graph-Conditioned Diffusion Models

• Graph-conditioned diffusion is a design paradigm where graph structures actively guide generative dynamics by serving as both the input and structural constraint.
• It spans multiple regimes—conditioning on graph objects, non-graph modalities, and graph-defined operators—enabling applications in counterfactual editing, molecular generation, and scene synthesis.
• Recent studies show that combining adaptive normalization, hard projection, and hybrid state spaces improves controllability and structural fidelity in diffusion-based models.

Graph-conditioned diffusion denotes a heterogeneous family of diffusion methods in which graph structure participates directly in the generative dynamics, the conditioning interface, or both. In current usage, the term spans at least three non-equivalent regimes: diffusion over graph objects conditioned on labels, properties, or instances; diffusion over non-graph modalities conditioned on graphs or graph-derived layouts; and graph-domain diffusion whose operator is itself defined by graph structure rather than by an external condition (Bechtoldt et al., 20 Nov 2025, Menneer et al., 3 Feb 2025, Stephenson et al., 3 Feb 2026). The resulting literature is unified less by a single formalism than by a common design principle: graph structure is treated as an explicit carrier of relational inductive bias during denoising, sampling, or control.

1. Conceptual scope and taxonomy

The expression “graph-conditioned diffusion” does not refer to one standardized model class. In the recent literature, it names several distinct conditioning loci.

Regime	Characteristic mechanism	Representative papers
Graph as the diffused object	Nodes and edges are noised and denoised, often under target-property conditions	(Bechtoldt et al., 20 Nov 2025, Liu et al., 2024)
Graph as an external condition	A graph, scene graph, or morphology graph conditions image or motion diffusion	(Farshad et al., 2023, Cao et al., 27 May 2025)
Graph as a constrained substrate	Sampling is projected to satisfy graph semantics or structural rules	(Prabhakar et al., 7 Jul 2025)
Graph-to-graph refinement	One graph is denoised or edited under another graph-derived signal	(Guo et al., 23 Dec 2025)

In graph-native generation, the conditioning variable is commonly a graph-level property, a target label, or a partially preserved input graph. “Graph Diffusion Counterfactual Explanation” generates counterfactual graphs by partially corrupting an observed graph and reverse-denoising it under a desired target outcome, so the graph is both the state being diffused and the instance being edited (Bechtoldt et al., 20 Nov 2025). “Graph Diffusion Transformers for Multi-Conditional Molecular Generation” instead treats graph conditioning as multi-property control for de novo molecular generation, with mixed numerical and categorical condition sets $\mathcal{C}=\{c_1,\dots,c_M\}$ and a target conditional distribution $p_\theta(G\mid\mathcal{C})$ (Liu et al., 2024).

In non-graph generation, graphs act as structured control signals. Scene-graph methods translate relational descriptions into layouts, tokens, or guidance terms for image synthesis rather than diffusing graphs themselves (Farshad et al., 2023). Motion retargeting across robot embodiments uses source and target morphology graphs, together with a partial joint correspondence map, as conditions for motion-sequence denoising (Cao et al., 27 May 2025).

A common misconception is that graph-conditioned diffusion always means class-conditional graph generation. The literature shows a broader picture. Some models condition on target properties, some on partially observed graphs, some on geometry or embodiment structure, and some use graph structure only to parameterize the forward or reverse operator. This suggests that “conditioning” in this area is best understood as an umbrella over graph-level semantics, graph-resident structure, and graph-derived constraints rather than a single probabilistic interface.

2. Representational choices and state spaces

A first axis of variation is the state space in which diffusion is performed. In discrete attributed graph models, graphs are typically written as $G=(X,E)$, where $X$ is a node-feature matrix and $E$ is an edge-type tensor. In the counterfactual framework of (Bechtoldt et al., 20 Nov 2025), $X\in\mathbb{R}^{n\times a}$ is encoded as one-hot categorical node types and $E\in\mathbb{R}^{n\times n\times b}$ as categorical edge types, so diffusion acts on node and edge categories directly. The graphs are dense-tensor representations with bounded size in each experimental setting.

Other works compress the graph state further. GraphGUIDE defines the graph as a binary edge-existence vector $x_t\in\{0,1\}^{\binom{n}{2}}$, so the diffusion process operates only on edge presence and every intermediate state remains a valid undirected graph (Tseng et al., 2023). CDGS also separates edge existence from edge type, but diffuses a continuous real-valued graph state and repeatedly decodes a binary adjacency $\bar{\mathbf{A}}_t$ from the current edge-existence channel for conditioning (Huang et al., 2023).

Graph DiT adopts a different discrete representation in which each node becomes a graph token containing both its own atom type and the flattened bond-type features to all $N$ nodes. The resulting matrix is $p_\theta(G\mid\mathcal{C})$0 with $p_\theta(G\mid\mathcal{C})$1, where $p_\theta(G\mid\mathcal{C})$2 is the node-type vocabulary size and $p_\theta(G\mid\mathcal{C})$3 includes non-bond as an edge type (Liu et al., 2024). This tokenization makes node and edge corruption jointly accessible to a Transformer denoiser.

Matrix-valued graph diffusion is another branch. “Generator-based Graph Generation via Heat Diffusion” diffuses the adjacency matrix itself through the heat semigroup

$p_\theta(G\mid\mathcal{C})$4

with $p_\theta(G\mid\mathcal{C})$5 in the main experiments, so the state lives in a continuous matrix space rather than a categorical graph space (Stephenson et al., 3 Feb 2026). The graph Laplacian $p_\theta(G\mid\mathcal{C})$6 functions as the infinitesimal generator of the forward process.

Hybrid state spaces appear in spatial biological graphs. “Semantically Consistent Discrete Diffusion for 3D Biological Graph Modeling” decomposes the graph as $p_\theta(G\mid\mathcal{C})$7, with $p_\theta(G\mid\mathcal{C})$8 for node coordinates and $p_\theta(G\mid\mathcal{C})$9 for one-hot edge labels, and then uses continuous Gaussian diffusion for coordinates plus discrete diffusion for labeled edges (Prabhakar et al., 7 Jul 2025). This mixed formulation is explicitly geometry-conditioned in its second stage.

When the target modality is not itself a graph, the representation often becomes heterogeneous. HIG models conditioning variables and image features as two interconnected graphs, with semantic nodes, image nodes, and typed links between them, so graph structure is present inside the denoiser’s intermediate representation rather than only at the input boundary (Menneer et al., 3 Feb 2025).

3. Conditioning regimes

Graph-conditioned diffusion can be organized by how and where the condition enters the model.

A standard regime is property-conditioned graph generation. Graph DiT embeds mixed-type conditions and injects them into the denoiser through adaptive layer normalization,

$G=(X,E)$0

with $G=(X,E)$1 as the combined condition representation (Liu et al., 2024). DiNAS uses graph-level conditions such as accuracy and latency classes, appends learned condition embeddings to node and edge features, and applies multi-conditioned classifier-free guidance during sampling (Asthana et al., 2024). GDCE conditions on a desired target outcome $G=(X,E)$2 after partial corruption of a specific input graph, so the same reverse model simultaneously expresses target steering, instance anchoring, and distributional realism (Bechtoldt et al., 20 Nov 2025).

A second regime is self-derived structural conditioning. CDGS does not rely on an external property label. Instead, it quantizes the current noisy edge state into a discrete adjacency $G=(X,E)$3 and conditions the denoiser as

$G=(X,E)$4

so reverse denoising in a continuous graph space is guided by a discrete structure decoded online from the current sample (Huang et al., 2023). In the 3D biological setting, edge denoising is conditioned on generated coordinates $G=(X,E)$5, while semantic and optional structural constraints are enforced by a projection operator that repairs or rejects invalid edge proposals during sampling (Prabhakar et al., 7 Jul 2025).

A third regime is hard structural control at sampling time. GraphGUIDE trains an unconditional Bernoulli diffusion model over binary edge vectors, but imposes user-specified structural constraints by editing each intermediate graph $G=(X,E)$6 before the next reverse step (Tseng et al., 2023). The condition is therefore procedural rather than parametric. New constraints do not require retraining because they are injected directly into the discrete reverse trajectory.

Graph-conditioned image synthesis shows several further variants. SceneGenie first converts a prompt into a scene graph, predicts bounding boxes and masks with SG2SEG, and then uses graph-derived CLIP and segmentation losses to perturb a pretrained latent diffusion sampler at inference time (Farshad et al., 2023). “Scene Graph Conditioning in Latent Diffusion” describes two related strategies: graph-to-layout conditioning via ControlNet and graph-to-token conditioning via Gated Self-Attention, both designed for sparse image–scene-graph supervision (Fundel, 2023). In histopathology, graph-conditioned diffusion replaces text embeddings with graph-derived object-level embeddings that encode class, morphology, and position, then feeds them through the text-conditioning pathway of a cascaded image diffusion model (Cechnicka et al., 8 Oct 2025).

Graph conditioning also appears in motion and anomaly modeling. G-DReaM conditions motion denoising on a reference motion, a source embodiment graph, a target embodiment graph, and a joint correspondence matrix $G=(X,E)$7, with graph information injected through typed relation matrices and multi-conditional cross-attention (Cao et al., 27 May 2025). GiCiSAD compresses a learned skeleton graph into a graph-level latent $G=(X,E)$8 derived from past frames and conditions future-sequence diffusion on that latent, so graph structure acts as a predictive summary rather than as a denoised object (Karami et al., 2024). HIG conditions image denoising through a heterogeneous image graph processed repeatedly inside an EDM2-ControlNet pipeline, so the condition is neither a flat token sequence nor a rasterized control image (Menneer et al., 3 Feb 2025).

These cases make clear that graph conditioning may be explicit, implicit, or procedural. It may act through condition embeddings, structural decoding, hard projection, or sampling-time guidance. The literature therefore treats graphs not only as inputs but also as dynamic organizational constraints on the denoising process.

4. Diffusion and sampling formulations

Discrete graph diffusion remains a central formulation. In GDCE, the forward process uses categorical transition matrices for node and edge states,

$G=(X,E)$9

with cumulative transitions $X$0, and the reverse model predicts clean node and edge categories which are converted into reverse posteriors (Bechtoldt et al., 20 Nov 2025). DiNAS follows the same discrete pattern for node operations and binary edges in neural architecture DAGs, but extends sampling with multi-conditioned classifier-free guidance. In its implementation, unconditional and conditional node/edge predictions are linearly combined as

$X$1

before sampling reverse transitions (Asthana et al., 2024).

Continuous SDE/ODE formulations remain important when edge and node states are embedded in Euclidean space. CDGS uses a VP-SDE over continuous graph variables and defines a graph probability-flow ODE

$X$2

which enables graph DPM-Solver variants in the few-step regime (Huang et al., 2023). This formulation is continuous even though the condition is discrete. The same tension between continuous state evolution and discrete graph semantics motivates its online quantization mechanism.

Not all graph-domain diffusion is score-based. The heat-kernel generator-matching model defines a closed-form semigroup

$X$3

with infinitesimal generator $X$4, then learns a reverse-time surrogate generator by minimizing a Bregman divergence, experimentally instantiated as Frobenius MSE (Stephenson et al., 3 Feb 2026). This is graph-domain diffusion with graph-structure-aware corruption, but not a standard conditional DDPM or score-SDE.

A different alternative is endpoint-conditioned bridge modeling. GruM constructs a mixture of endpoint-conditioned OU bridges and learns the posterior mean endpoint graph

$X$5

which enters the drift directly (Jo et al., 2023). The model is therefore conditioned on an inferred terminal graph rather than on an external label or graph prompt. This shifts learning from local denoising fields to explicit prediction of the graph that the process should end at.

Graph-conditioned refinement can also occur in adjacency-vector space. IGDMRec applies Gaussian diffusion to each column $X$6 of a semantic item graph and conditions denoising on the corresponding behavioral-graph column $X$7. It uses classifier-free guidance,

$X$8

but at inference initializes the reverse process from the semantic relation vector rather than from pure Gaussian noise in order to refine, rather than regenerate, graph structure (Guo et al., 23 Dec 2025).

Across these formulations, sampling need not begin from an isotropic prior. Counterfactual editing starts from a partially noised observed graph (Bechtoldt et al., 20 Nov 2025); recommendation-oriented graph denoising starts from an existing semantic graph column (Guo et al., 23 Dec 2025); projected biological generation starts from an empty graph over generated coordinates (Prabhakar et al., 7 Jul 2025). This suggests that graph-conditioned diffusion is as much about where the reverse chain begins as about how the denoiser is parameterized.

5. Architectural patterns and condition injection

Conditioning mechanisms vary substantially at the architectural level. In graph-native molecular generation, Graph DiT uses a Transformer denoiser over graph tokens and conditions every layer through AdaLN rather than through cross-attention or prompt tokens (Liu et al., 2024). DiNAS likewise uses a Graph Transformer backbone, but appends condition embeddings to node and edge features and augments node representations with sinusoidal positional encodings to preserve DAG order (Asthana et al., 2024).

CDGS illustrates a hybrid graph denoiser. Its Hybrid Message Passing Block combines local GINE message passing on the quantized discrete graph with edge-gated fully connected attention, then maps the resulting node features back to both node and edge predictions (Huang et al., 2023). The local branch exploits the decoded graph structure as a neighborhood mask, while the attention branch propagates global context independently of the current sparse adjacency. This split is explicitly motivated by the need to model both local node-edge dependency and global graph statistics.

When graphs condition image diffusion, the injection path is frequently inherited from pretrained text-to-image systems. HIG inserts a magnitude-preserving GNN into an EDM2-ControlNet branch. Image features are reshaped into image nodes, fused with a heterogeneous conditioning graph, processed by typed graph convolutions, and then mapped back to feature maps for residual injection into the denoiser (Menneer et al., 3 Feb 2025). Histopathology graph conditioning instead tokenizes segmented objects and their relations with a graph-aware transformer, then uses the resulting embeddings as replacements for text embeddings in a cascaded image diffusion pipeline (Cechnicka et al., 8 Oct 2025).

Other methods route graph information indirectly. SceneGenie does not modify the U-Net to consume graph tokens; instead it uses a separately trained graph-to-layout model and injects graph information through sampling-time gradients defined on boxes, masks, and CLIP-based region scores (Farshad et al., 2023). In motion retargeting, G-DReaM places embodiment graphs inside the denoiser itself through typed relation matrices $X$9, graph-conditioned spatial attention biases $E$0, and correspondence-masked cross-attention to reference motion tokens (Cao et al., 27 May 2025). GiCiSAD uses an STS-GCN denoiser conditioned on a graph-level latent $E$1, but the exact internal fusion operator for $E$2 is left unspecified in the paper text (Karami et al., 2024).

A recurring architectural pattern is reuse of pretrained conditioning interfaces. Graph embeddings are mapped into text-conditioning slots (Cechnicka et al., 8 Oct 2025), layout-control branches (Fundel, 2023), or ControlNet residual streams (Menneer et al., 3 Feb 2025). This design reduces the cost of building graph-native denoisers from scratch, but it also means that some systems are more precisely graph-guided diffusion samplers than end-to-end graph-conditioned denoisers.

6. Empirical domains and characteristic behaviors

Empirically, graph-conditioned diffusion has been used for counterfactual explanation, controllable molecular generation, constrained biological graph synthesis, image synthesis, recommendation, anomaly detection, motion retargeting, and neural architecture search. The observed gains are usually concentrated in controllability, structural faithfulness, or constraint satisfaction rather than in one universal metric.

Domain	Representative finding	Paper
Counterfactual graph editing	On planar graphs, GDCE at $E$3 reports Validity $E$4, Accuracy $E$5, Mean-GED $E$6, versus FreeGress $E$7, $E$8, $E$9	(Bechtoldt et al., 20 Nov 2025)
Hard structural control	Controlled generation in GraphGUIDE achieved the target property in $X\in\mathbb{R}^{n\times a}$0 of showcased cases; unconditional generation produced those properties only $X\in\mathbb{R}^{n\times a}$1, $X\in\mathbb{R}^{n\times a}$2, and $X\in\mathbb{R}^{n\times a}$3 of the time	(Tseng et al., 2023)
3D biological graphs	The semantically consistent projector achieved $X\in\mathbb{R}^{n\times a}$4 semantic validity on both Circle of Willis and ATM airway datasets	(Prabhakar et al., 7 Jul 2025)
Graph-conditioned image synthesis	HIG reports improvement from previous best FID $X\in\mathbb{R}^{n\times a}$5 to $X\in\mathbb{R}^{n\times a}$6 on Visual Genome layout-to-image and from $X\in\mathbb{R}^{n\times a}$7 to $X\in\mathbb{R}^{n\times a}$8 on COCO-stuff mask-to-image	(Menneer et al., 3 Feb 2025)
Multi-conditional molecular generation	Graph DiT reports polymer average MAE $X\in\mathbb{R}^{n\times a}$9 and categorical accuracies $E\in\mathbb{R}^{n\times n\times b}$0 on BACE, $E\in\mathbb{R}^{n\times n\times b}$1 on BBBP, and $E\in\mathbb{R}^{n\times n\times b}$2 on HIV	(Liu et al., 2024)
Neural architecture search	DiNAS samples architectures in less than $E\in\mathbb{R}^{n\times n\times b}$3 seconds per architecture	(Asthana et al., 2024)

The same pattern appears in recommendation. IGDMRec improves over competitive multimodal recommendation baselines on four datasets, and its ablations identify removal of behavioral conditioning as the largest degradation among tested variants. Its robustness studies under noisy or incomplete modalities further support the interpretation that conditioning a semantic item graph on behavioral co-occurrence acts as graph denoising rather than merely as auxiliary representation learning (Guo et al., 23 Dec 2025).

Motion applications show a related behavior. G-DReaM uses one graph-conditioned denoiser across multiple robot embodiments, including graph-only embodiments without paired motion datasets, and reports about 15 seconds to generate a single retargeted motion on an RTX 4090 (Cao et al., 27 May 2025). This does not establish universal cross-embodiment transfer, but it does show that graph-conditioned denoising can encode morphology-sensitive constraints without training a separate retargeter for every robot pair.

A broader empirical regularity is a control–validity trade-off. In GDCE, larger perturbation depth $E\in\mathbb{R}^{n\times n\times b}$4 improves target attainment but reduces similarity to the source instance (Bechtoldt et al., 20 Nov 2025). In GraphGUIDE, hard control is exact because it is enforced directly in edge space, but the framework is limited to properties that can be expressed as edge-level constraints (Tseng et al., 2023). In projected biological diffusion, semantic validity is guaranteed by construction at sampling time, but the projector intervenes only on a small fraction of edges, so learned generative structure still carries most of the burden (Prabhakar et al., 7 Jul 2025). Together these findings indicate that graph-conditioned diffusion often succeeds by combining learned priors with explicit structural correction rather than by relying on one mechanism alone.

7. Limitations, ambiguities, and open directions

The first limitation is terminological. The literature uses the same phrase for graph-native conditional generation, graph-guided image synthesis, graph-parameterized operators, and graph-refinement procedures. This ambiguity is visible in the contrast between Laplacian-conditioned heat diffusion, which is unconditional in the probabilistic sense (Stephenson et al., 3 Feb 2026), and scene-graph-conditioned latent diffusion, where the graph is an external semantic control signal (Fundel, 2023). A plausible implication is that future work will need a sharper vocabulary distinguishing the diffused state, the conditioning source, and the enforcement mechanism.

A second limitation is architectural incompleteness. Several papers clearly specify the conditioning concept but not the full backbone details. The histopathology graph-conditioned model describes graph construction and the text-conditioning replacement pathway, yet omits transformer depth, head count, graph embedding dimensionality, and diffusion schedule specifics in the provided text (Cechnicka et al., 8 Oct 2025). The latent-diffusion scene-graph paper explicitly reports that its proposed methods were not fully trained because of time and resource constraints, so its contribution is more an architectural study than a conclusive benchmark (Fundel, 2023). Generator-based heat diffusion also acknowledges that its flattened MLP implementation is not architecturally permutation-equivariant even though the mathematical generator respects graph symmetry (Stephenson et al., 3 Feb 2026).

A third limitation is scalability and bounded-size bias. Many graph-native models are demonstrated on fixed or bounded-size regimes: 8-node planar graphs, molecules up to 38 heavy atoms in ZINC, graph-token molecular models capped at 50 nodes, or dense pairwise edge modeling in 3D anatomy (Bechtoldt et al., 20 Nov 2025, Liu et al., 2024, Prabhakar et al., 7 Jul 2025). This does not preclude extension to larger graphs, but the available evidence is concentrated on settings where dense tensors, full attention, or flattened edge channels remain tractable.

A fourth limitation concerns the nature of control. Some methods provide soft probabilistic guidance, some hard projection, and some manual intervention. Each carries a different failure mode. GDCE can trade fidelity against validity through $E\in\mathbb{R}^{n\times n\times b}$5 but has no explicit minimality regularizer (Bechtoldt et al., 20 Nov 2025). GraphGUIDE offers interpretable exact control, but only when the desired property can be operationalized as edge-level edits (Tseng et al., 2023). This suggests that “controllability” in graph-conditioned diffusion is not a uniform property; it depends on whether the target is structural, semantic, continuous, or instance-specific.

Taken together, the literature indicates that graph-conditioned diffusion is better viewed as a design space than as a closed method family. The major axes are now visible: discrete versus continuous graph state spaces, external versus internal graph conditions, learned versus projected constraint enforcement, and graph-native versus graph-guided denoisers. The present diversity of formulations suggests that future convergence, if it occurs, will likely come from clearer taxonomies and from modular combinations of graph structure, multi-condition control, and explicit validity mechanisms rather than from a single dominant definition.