mlsanityData to 14 Oct 2026

Graph-Coupled Oscillator Networks

arXiv 2202.02296 ↗First posted 4 Feb 2022cs.LG math.DS stat.MLcited by 124

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Abstract (arXiv)

We propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear controlled and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e.g. convolutional or attentional) as the coupling function, from which a multi-layer deep neural network is built up via the dynamics of the proposed ODEs. We relate the oversmoothing problem, commonly encountered in GNNs, to the stability of steady states of the underlying ODE and show that zero-Dirichlet energy steady states are not stable for our proposed ODEs. This demonstrates that the proposed framework mitigates the oversmoothing problem. Moreover, we prove that GraphCON mitigates the exploding and vanishing gradients problem to facilitate training of deep multi-layer GNNs. Finally, we show that our approach offers competitive performance with respect to the state-of-the-art on a variety of graph-based learning tasks.

Key facts. Cited by 124 papers in the map (data through 14 Oct 2026). Builds on 39 of its 72 references that are in the map. First posted on arXiv 4 Feb 2022.

Builds on

39 of 72 references.

References · 39

  1. 2017Attention Is All You Needcited by 60,361
  2. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  3. 2017Graph Attention Networkscited by 8,923
  4. 2017Inductive Representation Learning on Large Graphscited by 6,320
  5. 2018How Powerful are Graph Neural Networks?cited by 4,393
  6. 2018Neural Ordinary Differential Equationscited by 3,779
  7. 2017Neural Message Passing for Quantum Chemistrycited by 3,487
  8. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,089
  9. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,576
  10. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  11. 2018Graph Convolutional Neural Networks for Web-Scale Recommender Systemscited by 1,138
  12. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,102
27 more
  1. 2018Pitfalls of Graph Neural Network Evaluationcited by 1,007
  2. 2018Representation Learning on Graphs with Jumping Knowledge Networkscited by 877
  3. 2020Simple and Deep Graph Convolutional Networkscited by 860
  4. 2020Geom-GCN: Geometric Graph Convolutional Networkscited by 826
  5. 2019DropEdge: Towards Deep Graph Convolutional Networks on Node Classificationcited by 680
  6. 2020Benchmarking Graph Neural Networkscited by 680
  7. 2020On the Bottleneck of Graph Neural Networks and its Practical Implicationscited by 631
  8. 2019Cluster-GCN: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networkscited by 541
  9. 2020Principal Neighbourhood Aggregation for Graph Netscited by 478
  10. 2021Understanding over-squashing and bottlenecks on graphs via curvaturecited by 478
  11. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  12. 2017Residual Gated Graph ConvNetscited by 346
  13. 2017Stable Architectures for Deep Neural Networkscited by 341
  14. 2019Revisiting Graph Neural Networks: All We Have is Low-Pass Filterscited by 261
  15. 2021GRAND: Graph Neural Diffusioncited by 258
  16. 2017SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernelscited by 161
  17. 2019Graph Neural Ordinary Differential Equationscited by 154
  18. 2017Predicting multicellular function through multi-layer tissue networkscited by 138
  19. 2020Directional Graph Networkscited by 133
  20. 2019Continuous Graph Neural Networkscited by 114
  21. 2021PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equationscited by 106
  22. 2021Beltrami Flow and Neural Diffusion on Graphscited by 84
  23. 2020Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependenciescited by 82
  24. 2018GeniePath: Graph Neural Networks with Adaptive Receptive Pathscited by 66
  25. 2021UnICORNN: A recurrent model for learning very long time dependenciescited by 56
  26. 2019Discrete and Continuous Deep Residual Learning Over Graphscited by 13
  27. 2020Probabilistic Numeric Convolutional Neural Networkscited by 10

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 10; 2023: 30; 2024: 36; 2025: 28; 2026: 20.

124 papers.

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  1. 2023A Survey on Oversmoothing in Graph Neural Networkscited by 247
  2. 2023On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topologycited by 166
  3. 2022Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvaturecited by 98
  4. 2022Anti-Symmetric DGN: a stable architecture for Deep Graph Networkscited by 60
  5. 2022Understanding convolution on graphs via energiescited by 59
  6. 2022GREAD: Graph Neural Reaction-Diffusion Networkscited by 54
  7. 2023How does over-squashing affect the power of GNNs?cited by 53
  8. 2022Gradient Gating for Deep Multi-Rate Learning on Graphscited by 49
  9. 2023Demystifying Oversmoothing in Attention-Based Graph Neural Networkscited by 49
  10. 2023A Fractional Graph Laplacian Approach to Oversmoothingcited by 45
  11. 2025On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learningcited by 43
  12. 2023Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvaturecited by 41
38 more
  1. 2024Oscillatory State-Space Modelscited by 39
  2. 2024Artificial Kuramoto Oscillatory Neuronscited by 38
  3. 2022ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networkscited by 31
  4. 2024Long Range Propagation on Continuous-Time Dynamic Graphscited by 28
  5. 2023Reversible and irreversible bracket-based dynamics for deep graph neural networkscited by 26
  6. 2023Graph Neural Convection-Diffusion with Heterophilycited by 26
  7. 2023From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyondcited by 26
  8. 2023Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approachcited by 24
  9. 2023Adaptive Message Passing: A General Framework to Mitigate Oversmoothing, Oversquashing, and Underreachingcited by 24
  10. 2023Over-Squashing in Graph Neural Networks: A Comprehensive surveycited by 23
  11. 2023Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challengescited by 22
  12. 2024Cayley Graph Propagationcited by 21
  13. 2022Improving Graph Neural Networks with Learnable Propagation Operatorscited by 19
  14. 2023Deep learning for dynamic graphs: models and benchmarkscited by 19
  15. 2025Oversmoothing, Oversquashing, Heterophily, Long-Range, and more: Demystifying Common Beliefs in Graph Machine Learningcited by 19
  16. 2023Rank Collapse Causes Over-Smoothing and Over-Correlation in Graph Neural Networkscited by 18
  17. 2024Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FRONDcited by 17
  18. 2023Neural Oscillators are Universalcited by 16
  19. 2024Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patchingcited by 16
  20. 2023From Coupled Oscillators to Graph Neural Networks: Reducing Over-smoothing via a Kuramoto Model-based Approachcited by 15
  21. 2025Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networkscited by 14
  22. 2024A Survey on Learning from Graphs with Heterophily: Recent Advances and Future Directionscited by 13
  23. 2023Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networkscited by 12
  24. 2025Return of ChebNet: Understanding and Improving an Overlooked GNN on Long Range Taskscited by 12
  25. 2023Leveraging Label Non-Uniformity for Node Classification in Graph Neural Networkscited by 11
  26. 2024Temporal Graph ODEs for Irregularly-Sampled Time Seriescited by 11
  27. 2024Non-convolutional Graph Neural Networkscited by 11
  28. 2022Generalized energy and gradient flow via graph frameletscited by 10
  29. 2023Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matricescited by 10
  30. 2025Message-Passing State-Space Models: Improving Graph Learning with Modern Sequence Modelingcited by 10
  31. 2025Can You Hear Me Now? A Benchmark for Long-Range Graph Propagationcited by 10
  32. 2024On The Temporal Domain of Differential Equation Inspired Graph Neural Networkscited by 9
  33. 2025Learning to Dissipate Energy in Oscillatory State-Space Modelscited by 9
  34. 2023Robust Graph Neural Networks via Unbiased Aggregationcited by 8
  35. 2024Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theorycited by 8
  36. 2024Unitary convolutions for learning on graphs and groupscited by 8
  37. 2023Temporal Graph Neural Networks for Irregular Datacited by 7
  38. 2024Distributed-Order Fractional Graph Operating Networkcited by 7

50 most cited of 124 All 124 (JSON)