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Advection-Diffusion on Graphs: A Bakry-Emery Laplacian for Spectral Graph Neural Networks

Pierre-Gabriel Berlureau, Ali Hariri, Victor Kawasaki-Borruat, Mia Zosso, Pierre Vandergheynst

Abstract

Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing. Existing remedies such as graph transformers or rewiring typically incur high computational cost or require altering the graph structure. We introduce a Bakry-Emery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential, inducing task-dependent propagation dynamics without modifying topology. This operator has a well-behaved spectral decomposition and acts as a drop-in replacement for standard Laplacians in spectral GNNs. Building on this insight, we develop mu-ChebNet, a spectral architecture that jointly learns the potential and Chebyshev filters, effectively bridging message-passing adaptivity and spectral efficiency. Our theoretical analysis shows how the potential modulates the spectrum, enabling control of key graph properties. Empirically, mu-ChebNet delivers consistent gains on synthetic long-range reasoning tasks, as well as real-world benchmarks, while offering an interpretable routing field that reveals how information flows through the graph. This establishes the Bakry-Emery Laplacian as a principled and efficient foundation for adaptive spectral graph learning.

Advection-Diffusion on Graphs: A Bakry-Emery Laplacian for Spectral Graph Neural Networks

Abstract

Graph Neural Networks (GNNs) often struggle to propagate information across long distances due to oversmoothing and oversquashing. Existing remedies such as graph transformers or rewiring typically incur high computational cost or require altering the graph structure. We introduce a Bakry-Emery graph Laplacian that integrates diffusion and advection through a learnable node-wise potential, inducing task-dependent propagation dynamics without modifying topology. This operator has a well-behaved spectral decomposition and acts as a drop-in replacement for standard Laplacians in spectral GNNs. Building on this insight, we develop mu-ChebNet, a spectral architecture that jointly learns the potential and Chebyshev filters, effectively bridging message-passing adaptivity and spectral efficiency. Our theoretical analysis shows how the potential modulates the spectrum, enabling control of key graph properties. Empirically, mu-ChebNet delivers consistent gains on synthetic long-range reasoning tasks, as well as real-world benchmarks, while offering an interpretable routing field that reveals how information flows through the graph. This establishes the Bakry-Emery Laplacian as a principled and efficient foundation for adaptive spectral graph learning.
Paper Structure (28 sections, 6 theorems, 63 equations, 3 figures, 4 tables)

This paper contains 28 sections, 6 theorems, 63 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Related Work
  5. Background
  6. Graph Laplacians
  7. ChebNet
  8. Advection–diffusion
  9. Method
  10. The Bakry-Emery Laplacian on $\mathbb{R}^d$
  11. The Bakry-Emery Laplacian on Graphs
  12. Properties of the BE-Laplacian on Graphs
  13. A novel class of spectral GNNs
  14. Experiments
  15. Conclusion
  16. ...and 13 more sections

Key Result

Proposition 4.1

Let $\Omega \subset \mathbb{R}^d$ be a smooth domain and let $\mu(x) = Z^{-1} e^{-V(x)}$ be a strictly positive, integrable density on $\Omega$, with $V \in C^1(\Omega)$. Define the weighted Dirichlet form Then $\mathcal{D}_\mu$ admits the generator $L_\mu$ given by

Figures (3)

  • Figure 1: Standard vs. $\mu$-Laplacian on a ring graph. (a) Heat diffusion under the standard Laplacian produces nearly isotropic propagation. (b) The $\mu$-Laplacian induces an advection--diffusion geometry via a node-wise potential, biasing information flow without altering graph topology. The potential breaks the symmetry, resulting in anisotropic diffusion
  • Figure 2: Eigenvalue distribution comparison on the Barbell task using the standard normalized Laplacians vs $\mu-$Laplacian
  • Figure 3: Ring graph configuration. (Left) The query node is in purple and the answer node in yellow. Nodes on the rightward path contain noisy features. (Right) The resulting learned potential has broken the symmetry of the configuration, routing information on the most informative path.

Theorems & Definitions (13)

  • Proposition 4.1: Bakry--Émery operator as weighted diffusion
  • Proposition 4.2: Closed form of the graph Bakry--Émery Laplacian
  • Definition 4.3
  • Lemma 4.4
  • Theorem 4.5
  • Corollary 4.6
  • Corollary 4.7
  • proof
  • proof
  • proof
  • ...and 3 more