mlsanityData to 7 Oct 2026

A Fractional Graph Laplacian Approach to Oversmoothing

arXiv 2305.13084 ↗First posted 22 May 2023cs.LGcited by 45

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Abstract (arXiv)

Graph neural networks (GNNs) have shown state-of-the-art performances in various applications. However, GNNs often struggle to capture long-range dependencies in graphs due to oversmoothing. In this paper, we generalize the concept of oversmoothing from undirected to directed graphs. To this aim, we extend the notion of Dirichlet energy by considering a directed symmetrically normalized Laplacian. As vanilla graph convolutional networks are prone to oversmooth, we adopt a neural graph ODE framework. Specifically, we propose fractional graph Laplacian neural ODEs, which describe non-local dynamics. We prove that our approach allows propagating information between distant nodes while maintaining a low probability of long-distance jumps. Moreover, we show that our method is more flexible with respect to the convergence of the graph's Dirichlet energy, thereby mitigating oversmoothing. We conduct extensive experiments on synthetic and real-world graphs, both directed and undirected, demonstrating our method's versatility across diverse graph homophily levels. Our code is available at https://github.com/RPaolino/fLode .

Key facts. Cited by 45 papers in the map (data through 7 Oct 2026). Builds on 53 of its 72 references that are in the map. First posted on arXiv 22 May 2023.

Builds on

53 of 72 references.

References · 53

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38 more
  1. 2018Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learningcited by 1,099
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  3. 2018Predict then Propagate: Graph Neural Networks meet Personalized PageRankcited by 987
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  8. 2020Adaptive Universal Generalized PageRank Graph Neural Networkcited by 592
  9. 2019Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological Viewcited by 544
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  15. 2023A critical look at the evaluation of GNNs under heterophily: Are we really making progress?cited by 332
  16. 2019PairNorm: Tackling Oversmoothing in GNNscited by 297
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  18. 2020A Note on Over-Smoothing for Graph Neural Networkscited by 245
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  21. 2021Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networkscited by 182
  22. 2022Finding Global Homophily in Graph Neural Networks When Meeting Heterophilycited by 179
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 6; 2024: 12; 2025: 21; 2026: 6.

45 papers.

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  1. 2025On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learningcited by 42
  2. 2023From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyondcited by 26
  3. 2023Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challengescited by 22
  4. 2023Rank Collapse Causes Over-Smoothing and Over-Correlation in Graph Neural Networkscited by 18
  5. 2025Towards Quantifying Long-Range Interactions in Graph Machine Learning: a Large Graph Dataset and a Measurementcited by 18
  6. 2025Oversmoothing, Oversquashing, Heterophily, Long-Range, and more: Demystifying Common Beliefs in Graph Machine Learningcited by 18
  7. 2024Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FRONDcited by 16
  8. 2023Feature Transportation Improves Graph Neural Networkscited by 15
  9. 2025Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networkscited by 14
  10. 2025Return of ChebNet: Understanding and Improving an Overlooked GNN on Long Range Taskscited by 12
  11. 2023Graph Convolutions Enrich the Self-Attention in Transformers!cited by 11
  12. 2024On The Temporal Domain of Differential Equation Inspired Graph Neural Networkscited by 9
33 more
  1. 2024Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theorycited by 8
  2. 2024Unitary convolutions for learning on graphs and groupscited by 8
  3. 2025Are We Measuring Oversmoothing in Graph Neural Networks Correctly?cited by 6
  4. 2025Improving the Effective Receptive Field of Message-Passing Neural Networkscited by 6
  5. 2025What makes a good feedforward computational graph?cited by 5
  6. 2023Hetero2Net: Heterophily-aware Representation Learning on Heterogenerous Graphscited by 4
  7. 2024Simplifying the Theory on Over-Smoothingcited by 4
  8. 2024Rethinking Spectral Graph Neural Networks with Spatially Adaptive Filteringcited by 3
  9. 2024Foundations and Frontiers of Graph Learning Theorycited by 3
  10. 2024Improving Graph Neural Networks by Learning Continuous Edge Directionscited by 3
  11. 2025Learning Advanced Self-Attention for Linear Transformers in the Singular Value Domaincited by 3
  12. 2024Gradient Flow of Energy: A General and Efficient Approach for Entity Alignment Decodingcited by 2
  13. 2025Efficient Learning on Large Graphs using a Densifying Regularity Lemmacited by 2
  14. 2025Partition-wise Graph Filtering: A Unified Perspective Through the Lens of Graph Coarseningcited by 2
  15. 2024Dirac--Bianconi Graph Neural Networks -- Enabling Non-Diffusive Long-Range Graph Predictionscited by 1
  16. 2025Piecewise Constant Spectral Graph Neural Networkcited by 1
  17. 2025MatRL: Provably Generalizable Iterative Algorithm Discovery via Monte-Carlo Tree Searchcited by 1
  18. 2025Measuring Over-smoothing beyond Dirichlet energycited by 1
  19. 2026Nonlinear Laplacians Improve Signed-Directed Graph Learningcited by 1
  20. 2024Understanding and Guiding Weakly Supervised Entity Alignment with Potential Isomorphism Propagationcited by 0
  21. 2024Information propagation dynamics in Deep Graph Networkscited by 0
  22. 2025Rethinking Link Prediction for Directed Graphscited by 0
  23. 2025Simplifying Graph Convolutional Networks with Redundancy-Free Neighborscited by 0
  24. 2025Simple Graph Contrastive Learning via Fractional-order Neural Diffusion Networkscited by 0
  25. 2025TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flowscited by 0
  26. 2025Non-Dissipative Graph Propagation for Non-Local Community Detectioncited by 0
  27. 2025How Wide and How Deep? Mitigating Over-Squashing of GNNs via Channel Capacity Constrained Estimationcited by 0
  28. 2025Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusioncited by 0
  29. 2026Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNscited by 0
  30. 2026Graph Negative Feedback Bias Correction Framework for Adaptive Heterophily Modelingcited by 0
  31. 2026Latent-Hysteresis Graph ODEs: Modeling Coupled Topology-Feature Evolution via Continuous Phase Transitionscited by 0
  32. 2026Remedying Coarsening-Based GNN Training under Heterophily via Adaptive Complementary Enhancementcited by 0
  33. 2026"very likely" Means "uncertain"? How LLMs Diverge from Humans in Linguistic Uncertainty Quantificationcited by 0