Graph Diffusion

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Graph diffusion describes processes that spread signals, labels, or probability across connected nodes, and it underpins both graph learning and generation of graph-structured data. Methods include spectral heat kernels and random walks, neural diffusion operators such as personalized PageRank and sheaf-based propagation, and continuous or discrete denoising models such as score-based diffusion and DiGress-style methods. These approaches are used for node classification and clustering, molecular and network generation, forecasting on sensor graphs, and prediction on irregular physical meshes. A central tension is that stronger or longer-range propagation can capture useful global structure but also cause oversmoothing, while generative models must balance fidelity, sampling cost, structural constraints, and risks such as memorization.

Latest papers

75 papers14 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

Apr 13, 2026

Learning Discrete Diffusion of Graphs via Free-Energy Gradient Flows

Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (

{W}_2

) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the increasing popularity of diffusion models on discrete spaces using continuous-time Markov chains, a parallel theoretical framework based on gradient flows has remained elusive due to intrinsic challenges in translating the

{W}_2

distance directly into these settings. In this work, we propose the first computational approach addressing these challenges, leveraging an appropriate metric

W_K

on the simplex of probability distributions, which enables us to interpret widely used discrete diffusion paths, such as the discrete heat equation, as gradient flows of specific free-energy functionals. Through this theoretical insight, we introduce a novel methodology for learning diffusion dynamics over discrete spaces, which recovers the underlying functional directly by leveraging first-order optimality conditions for the JKO scheme. The resulting method optimizes a simple quadratic loss, trains extremely fast, does not require individual sample trajectories, and only needs a numerical preprocessing computing

W_K

-geodesics. We validate our method through extensive numerical experiments on synthetic data, showing that we can recover the underlying functional for a variety of graph classes.

Dario Rancati, Jan Maas, Francesco Locatello

Institute of Science and Technology Austria

Diffusion Models Graph Diffusion Wasserstein Gradient Flows

Apr 2, 2026

Bias mitigation in graph diffusion models

Most existing graph diffusion models have significant bias problems. We observe that the forward diffusion's maximum perturbation distribution in most models deviates from the standard Gaussian distribution, while reverse sampling consistently starts from a standard Gaussian distribution, which results in a reverse-starting bias. Together with the inherent exposure bias of diffusion models, this results in degraded generation quality. This paper proposes a comprehensive approach to mitigate both biases. To mitigate reverse-starting bias, we employ a newly designed Langevin sampling algorithm to align with the forward maximum perturbation distribution, establishing a new reverse-starting point. To address the exposure bias, we introduce a score correction mechanism based on a newly defined score difference. Our approach, which requires no network modifications, is validated across multiple models, datasets, and tasks, achieving state-of-the-art results.Code is at https://github.com/kunzhan/spp

Meng Yu, Kun Zhan

School of Information Science & Engineering, Lanzhou University

Diffusion Models Graph Diffusion Diffusion Model Sampling Langevin Dynamics

Mar 16, 2026

PiGRAND: Physics-informed Graph Neural Diffusion for Intelligent Additive Manufacturing

A comprehensive understanding of heat transport is essential for optimizing various mechanical and engineering applications, including 3D printing. Recent advances in machine learning, combined with physics-based models, have enabled a powerful fusion of numerical methods and data-driven algorithms. This progress is driven by the availability of limited sensor data in various engineering and scientific domains, where the cost of data collection and the inaccessibility of certain measurements are high. To this end, we present PiGRAND, a Physics-informed graph neural diffusion framework. In order to reduce the computational complexity of graph learning, an efficient graph construction procedure was developed. Our approach is inspired by the explicit Euler and implicit Crank-Nicolson methods for modeling continuous heat transport, leveraging sub-learning models to secure the accurate diffusion across graph nodes. To enhance computational performance, our approach is combined with efficient transfer learning. We evaluate PiGRAND on thermal images from 3D printing, demonstrating significant improvements in prediction accuracy and computational performance compared to traditional graph neural diffusion (GRAND) and physics-informed neural networks (PINNs). These enhancements are attributed to the incorporation of physical principles derived from the theoretical study of partial differential equations (PDEs) into the learning model. The PiGRAND code is open-sourced on GitHub: https://github.com/bu32loxa/PiGRAND

Benjamin Uhrich, Tim Häntschel, Erhard Rahm

Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany. · Leipzig University, Leipzig, Germany.

Physics-Informed GNNs Graph Diffusion Additive Manufacturing

Mar 16, 2026

Tackling Over-smoothing on Hypergraphs: A Ricci Flow-guided Neural Diffusion Approach

Hypergraph neural networks (HGNNs) have demonstrated strong capabilities in modeling complex higher-order relationships. However, existing HGNNs often suffer from over-smoothing as the number of layers increases and lack effective control over message passing among nodes. Inspired by the theory of Ricci flow in differential geometry, we theoretically establish that introducing discrete Ricci flow into hypergraph structures can effectively regulate node feature evolution and thereby alleviate over-smoothing. Building on this insight, we propose Ricci Flow-guided Hypergraph Neural Diffusion(RFHND), a novel message passing paradigm for hypergraphs guided by discrete Ricci flow. Specifically, RFHND is based on a PDE system that describes the continuous evolution of node features on hypergraphs and adaptively regulates the rate of information diffusion at the geometric level, preventing feature homogenization and producing high-quality node representations. Experimental results show that RFHND significantly outperforms existing methods across multiple benchmark datasets and demonstrates strong robustness, while also effectively mitigating over-smoothing.

Mengyao Zhou, Zhiheng Zhou, Xiao Han +3

the Academy of Mathematics and Systems Science, Chinese Academy of Sciences and also with the University of Chinese Academy of Sciences, Beijing 100190, China · the School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China · School of Artificial Intelligence, Beihang University, Beijing 100191, China +1

Over-Smoothing in GNNs Hypergraph Neural Networks Hypergraph Learning GNN Oversmoothing Graph Neural Networks Graph Diffusion

Mar 15, 2026

MorphSNN: Adaptive Graph Diffusion and Structural Plasticity for Spiking Neural Networks

Spiking Neural Networks (SNNs) currently face a critical bottleneck: while individual neurons exhibit dynamic biological properties, their macro-scopic architectures remain confined within conventional connectivity patterns that are static and hierarchical. This discrepancy between neuron-level dynamics and network-level fixed connectivity eliminates critical brain-like lateral interactions, limiting adaptability in changing environments. To address this, we propose MorphSNN, a backbone framework inspired by biological non-synaptic diffusion and structural plasticity. Specifically, we introduce a Graph Diffusion (GD)mechanism to facilitate efficient undirected signal propagation, complementing the feedforward hierarchy. Furthermore, it incorporates a Spatio-Temporal Structural Plasticity (STSP) mechanism, endowing the network with the capability for instance-specific, dynamic topological reorganization, thereby overcoming the limitations of fixed topologies. Experiments demonstrate that MorphSNN achieves state-of-the-art accuracy on static and neuromorphic datasets; for instance, it reaches 83.35% accuracy on N-Caltech101 with only 5 timesteps. More importantly, its self-evolving topology functions as an intrinsic distribution fingerprint, enabling superior Out-of- Distribution (OOD) detection without auxiliary training. The code is available at anonymous.4open.science/r/MorphSNN-B0BC.

Yongsheng Huang, Peibo Duan, Yujie Wu +7

School of Software, Northeastern University, China · Guangdong Institute of Intelligence Science and Technolog, China · The Hong Kong Polytechnic University, China +1

Spiking Neural Networks Dynamic Neural Networks Graph Diffusion

Mar 9, 2026

Semantic Level of Detail for Knowledge Graphs: Discovering Abstraction Boundaries via Spectral Heat Diffusion

Graph-structured knowledge systems -- from knowledge graphs to GraphRAG pipelines -- organize information into hierarchical communities, yet lack a principled mechanism for continuous resolution control: where do the qualitative boundaries between abstraction levels lie, and how should an agent navigate them? Current approaches rely on discrete community detection with manually tuned resolution parameters (e.g., Leiden

γ

), offering no continuous zoom and no formal guarantees. We introduce Semantic Level of Detail (SLoD), a framework that addresses both problems by defining a continuous zoom operator via heat kernel diffusion on a graph Laplacian whose kNN structure is induced by a Poincare-ball embedding. We prove hierarchical coherence in the tree limit (exact tree with Sarkar embedding), with bounded approximation error, and demonstrate consistent boundary-detection behaviour on noisy hierarchies; spectral gaps in the graph Laplacian then induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- detectable without manual resolution tuning. On synthetic hierarchies (HSBM, 1024 nodes), spectral clustering at the BoundaryScan-detected scale recovers planted levels, with macro ARI saturating at 1.00 in the high-SNR regime (50-seed median) and meso ARI reaching 0.89 [0.86, 0.92] at r=200. On the full WordNet noun hierarchy (82K synsets), using 100 stratified leaf queries, detected boundaries align with true taxonomic depth (

τ= 0.79

), demonstrating meaningful abstraction-level discovery in real-world knowledge graphs without resolution-parameter tuning. The composite weights, MAD threshold, and kNN-parameter rule (

k = \max(10, \min(\lfloor\sqrt{N}\rfloor, 50))

) use defaults that transferred unchanged between HSBM and WordNet; their behaviour on graphs with implicit or qualitatively different hierarchical structure is open.

Edward Izgorodin

Mnemoverse.AI, Funchal, Madeira, Portugal

Knowledge Graphs Graph Clustering Graph Diffusion

Mar 2, 2026

Mitigating topology biases in Graph Diffusion via Counterfactual Intervention

Graph diffusion models have gained significant attention in graph generation tasks, but they often inherit and amplify topology biases from sensitive attributes (e.g. gender, age, region), leading to unfair synthetic graphs. Existing fair graph generation using diffusion models is limited to specific graph-based applications with complete labels or requires simultaneous updates for graph structure and node attributes, making them unsuitable for general usage. To relax these limitations by applying the debiasing method directly on graph topology, we propose Fair Graph Diffusion Model (FairGDiff), a counterfactual-based one-step solution that mitigates topology biases while balancing fairness and utility. In detail, we construct a causal model to capture the relationship between sensitive attributes, biased link formation, and the generated graph structure. By answering the counterfactual question "Would the graph structure change if the sensitive attribute were different?", we estimate an unbiased treatment and incorporate it into the diffusion process. FairGDiff integrates counterfactual learning into both forward diffusion and backward denoising, ensuring that the generated graphs are independent of sensitive attributes while preserving structural integrity. Extensive experiments on real-world datasets demonstrate that FairGDiff achieves a superior trade-off between fairness and utility, outperforming existing fair graph generation methods while maintaining scalability.

Wendi Wang, Jiaxi Yang, Yongkang Du +1

Purdue University · Pennsylvania State University

Graph Generation Graph Diffusion Counterfactual Fairness

Feb 22, 2026

DGPO: RL-Steered Graph Diffusion for Neural Architecture Generation

Reinforcement learning fine-tuning has proven effective for steering generative diffusion models toward desired properties in image and molecular domains. Graph diffusion models have similarly been applied to combinatorial structure generation, including neural architecture search (NAS). However, neural architectures are directed acyclic graphs (DAGs) where edge direction encodes functional semantics such as data flow-information that existing graph diffusion methods, designed for undirected structures, discard. We propose Directed Graph Policy Optimization (DGPO), which extends reinforcement learning fine-tuning of discrete graph diffusion models to DAGs via topological node ordering and positional encoding. Validated on NAS-Bench-101 and NAS-Bench-201, DGPO matches the benchmark optimum on all three NAS-Bench-201 tasks (91.61%, 73.49%, 46.77%). The central finding is that the model learns transferable structural priors: pretrained on only 7% of the search space, it generates near-oracle architectures after fine-tuning, within 0.32 percentage points of the full-data model and extrapolating 7.3 percentage points beyond its training ceiling. Bidirectional control experiments confirm genuine reward-driven steering, with inverse optimization reaching near random-chance accuracy (9.5%). These results demonstrate that reinforcement learning-steered discrete diffusion, once extended to handle directionality, provides a controllable generative framework for directed combinatorial structures.

Aleksei Liuliakov, Luca Hermes, Barbara Hammer

Machine Learning Group, Bielefeld University, Germany

Graph Generation Graph Diffusion Neural Architecture Search Diffusion Model Fine-Tuning RL for Diffusion Models

Feb 20, 2026

Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNs

Long-range graph tasks are challenging for Graph Neural Networks (GNNs): global mechanisms such as attention or rewiring schemes can be computationally expensive, while deep local propagation is prone to vanishing gradients, oversmoothing, and oversquashing. The introduced mu-ChebNet architecture is a simple spectral GNN that learns a node-wise weight function mu before applying ChebNet-style filters. The learned weighting mu induces a modified graph Laplacian which effectively changes the propagation geometry without altering the graph topology. This task-dependent geometry promotes preferred routes for information propagation, thereby helping long-range signals avoid highly contractive bottlenecks, and obviating the need for repeated layer stacking. In practice, we replace the fixed graph Laplacian L by a learned operator L_mu, keeping the proposed mu-ChebNet architecture lightweight while making propagation task-adaptive. Furthermore, we provide a spectral analysis demonstrating how mu modulates propagation dynamics, and empirically observe improved performance on both synthetic long-range reasoning tasks and real-world graph benchmarks. The learned weight function is not only interpretable, but also offers a lightweight alternative to attention and rewiring for adaptive graph propagation.

Mia Zosso, Ali Hariri, Victor Kawasaki-Borruat +2

École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland · École Normale Supérieure – PSL, Paris, France

Long-Range Graph Learning Graph Representation Learning Spectral Graph Convolution Graph Diffusion

Feb 19, 2026

MolHIT: Advancing Molecular-Graph Generation with Hierarchical Discrete Diffusion Models

Molecular generation with diffusion models has emerged as a promising direction for AI-driven drug discovery and materials science. While graph diffusion models have been widely adopted due to the discrete nature of 2D molecular graphs, existing models suffer from low chemical validity and struggle to meet the desired properties compared to 1D modeling. In this work, we introduce MolHIT, a powerful molecular graph generation framework that overcomes long-standing performance limitations in existing methods. MolHIT is based on the Hierarchical Discrete Diffusion Model, which generalizes discrete diffusion to additional categories that encode chemical priors, and decoupled atom encoding that splits the atom types according to their chemical roles. Overall, MolHIT achieves new state-of-the-art performance on the MOSES dataset with near-perfect validity for the first time in graph diffusion, surpassing strong 1D baselines across multiple metrics. We further demonstrate strong performance in downstream tasks, including multi-property guided generation and scaffold extension.

Hojung Jung, Rodrigo Hormazabal, Jaehyeong Jo +5

KAIST AI · LG AI Research · Seoul National University

Molecular Graph Generation Diffusion Models Graph Diffusion

Dec 20, 2025

Out-of-Distribution Detection in Molecular Complexes via Diffusion Models for Irregular Graphs

Predictive machine learning models generally excel on in-distribution data, but their performance degrades on out-of-distribution (OOD) inputs. Reliable deployment therefore requires robust OOD detection, yet this is particularly challenging for irregular 3D graphs that combine continuous geometry with categorical identities and are unordered by construction. Here, we present a probabilistic OOD detection framework for complex 3D graph data built on a diffusion model that learns a density of the training distribution in a fully unsupervised manner. A key ingredient we introduce is a unified continuous diffusion over both 3D coordinates and discrete features: categorical identities are embedded in a continuous space and trained with cross-entropy, while the corresponding diffusion score is obtained analytically via posterior-mean interpolation from predicted class probabilities. This yields a single self-consistent probability-flow ODE (PF-ODE) that produces per-sample log-likelihoods, providing a principled typicality score for distribution shift. We validate the approach on protein-ligand complexes and construct strict OOD datasets by withholding entire protein families from training. PF-ODE likelihoods identify held-out families as OOD and correlate strongly with prediction errors of an independent binding-affinity model (GEMS), enabling a priori reliability estimates on new complexes. Beyond scalar likelihoods, we show that multi-scale PF-ODE trajectory statistics - including path tortuosity, flow stiffness, and vector-field instability - provide complementary OOD information. Modeling the joint distribution of these trajectory features yields a practical, high-sensitivity detector that improves separation over likelihood-only baselines, offering a label-free OOD quantification workflow for geometric deep learning.

David Graber, Victor Armegioiu, Rebecca Buller +1

Seminar for Applied Mathematics, Department of Mathematics and ETH AI Center, ETH Zurich, 8092 Zurich, Switzerland · Institute for Computational Life Sciences, Zurich University of Applied Sciences, 8820 Waedenswil, Switzerland · Department of Chemistry, Biochemistry and Pharmaceutical Sciences, University of Bern, 3012 Bern, Switzerland

Diffusion-Based OOD Detection OOD Detection Graph Diffusion

Oct 6, 2025

Fractional Heat Kernel for Semi-Supervised Graph Learning with Small Training Sample Size

We develop a source-driven fractional heat-kernel framework for semi-super-vised graph learning that combines nonlocal propagation with sustained label information. A fixed nonzero label source compatible with the Laplacian null space prevents asymptotic collapse into that space, providing a mechanism for mitigating oversmoothing at long diffusion times. The fractional order controls the relative modal attenuation and the spectral weighting of the sustained response, while the diffusion time sets the propagation horizon. We characterize conservation laws and equilibria on normalized and disconnected graphs, develop a null-space deflation, and analyze the approximation of the propagators. On Two-Moon, fractional orders improve source-free propagation, while compatible source-driven diffusion exceeds

96\%

mean accuracy with one training label per class at orders

0.8

and

1

. On Cora and CiteSeer, the source-driven pipeline improves mean accuracy over GAT by

9.2

and

8.0

percentage points at one training label per class, with model selection on

500

labeled validation nodes and closely comparable classical and fractional pipeline configurations; it matches GAT on PubMed and trails it at ten and twenty labels per class on Cora. Within GraphHeat, validation-based exponent selection at a common diffusion time yields a paired gain of

1.45

percentage points at one label per class.

Farid Bozorgnia, Vyacheslav Kungurtsev, Shirali Kadyrov +1

Over-Smoothing in GNNs Graph-Based Semi-Supervised Learning Graph-Based Label Propagation Graph Diffusion Graph Neural Networks

Jun 29, 2025

Double-Diffusion: Balancing Speed, Accuracy, and Uncertainty in Probabilistic Forecasting for Urban Sensor Networks

Urban sensor networks need forecasts that are accurate, carry useful uncertainty, and refresh fast enough to act on as new readings arrive. These goals conflict: deterministic models give no distribution, while diffusion forecasters model uncertainty but denoise from pure noise over many steps. We present Double-Diffusion, which integrates a closed-form graph-heat prior into a denoising diffusion model. The prior is a parameter-free low-pass forecast over the sensor graph, and it serves two roles: it is the residual target the model generates, and it conditions the denoiser. The reverse process therefore starts near the prior and denoises a short warm-started chain instead of synthesizing from pure noise; the name denotes this composition, a graph diffusion feeding a denoising diffusion. A compact denoiser, DD-Net, is trained as a Denoising Diffusion Probabilistic Model (DDPM) in the Resfusion warm-start formulation, so generation refines the prior over a short truncated chain rather than synthesizing from pure noise; a graph-spectral read-out of the prior residual sets its switchable spatial filter per domain from training data alone. On four real-world air quality and traffic networks, Double-Diffusion attains the best CRPS of all probabilistic methods on every dataset and stays competitive in point accuracy with the strongest baselines, at a fraction of the sampling cost of from-noise diffusion. The code is available at: https://github.com/teddyicare/Double-Diffusion

Hanlin Dong, Arian Prabowo, Hao Xue +4

University of New South Wales Sydney, Australia · University of Maryland College Park, United States · The Hong Kong University of Science and Technology (Guangzhou) Guangzhou, China

Spatiotemporal Forecasting Diffusion Model Inference Acceleration Probabilistic Time Series Forecasting Diffusion Models Graph Diffusion

Mar 8, 2025

Backdoor Attacks on Discrete Graph Diffusion Models

Diffusion models have demonstrated remarkable generative capabilities in continuous data domains such as images and videos. Recently, discrete graph diffusion models (DGDMs) have extended this success to graph generation, achieving state-of-the-art performance. However, deploying DGDMs in safety-critical applications, such as drug discovery, poses significant risks without a thorough understanding of their security vulnerabilities. In this work, we conduct the first study of backdoor attacks on DGDMs, a potent threat that manipulates both the training and generation phases of graph diffusion. We begin by formalizing the threat model and then design a backdoor attack that enables the compromised model to: 1) generate high-quality, benign graphs when the backdoor is not activated, 2) produce effective, stealthy, and persistent backdoored graphs when triggered, and 3) preserve fundamental graph properties (permutation equivariance and exchangeability) even under attack. We validate 1) and 2) empirically, both with and without backdoor defenses, and support 3) through theoretical analysis inspired by prior work.

Jiawen Wang, Samin Bin Karim, Yuan Hong +1

Department of Computer Science, Illinois Institute of Technology · School of Computing, University of Connecticut

Graph Generation Backdoor Attacks Graph Diffusion

Oct 1, 2022

Parametrized Power-Iteration Clustering for Directed Graphs

Vertex-level clustering for directed graphs (digraphs) remains challenging as edge directionality breaks the key assumptions underlying popular spectral methods, which also incur the overhead of eigen-decomposition. This paper proposes Parametrized Power-Iteration Clustering (ParPIC), a random-walk-based clustering method for weakly connected digraphs. This builds over the Power-Iteration Clustering paradigm, which uses the rows of the iterated diffusion operator as a data embedding. ParPIC has three important features: the use of parametrized reversible random walk operators, the automatic tuning of the diffusion time, and the efficient truncation of the final embedding, which produces low-dimensional data representations and reduces complexity. Empirical results on synthetic and real-world graphs demonstrate that ParPIC achieves competitive clustering accuracy with improved scalability relative to spectral and teleportation-based methods.

Gwendal Debaussart-Joniec, Harry Sevi, Matthieu Jonckheere +1

Universit´e Paris-Saclay, ENS Paris-Saclay, Centre Borelli, CNRS, France · CNRS, LAAS, France

Graph Clustering Graph Embedding Graph Diffusion