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Graph Adversarial Diffusion Convolution

Songtao Liu, Jinghui Chen, Tianfan Fu, Lu Lin, Marinka Zitnik, Dinghao Wu

TL;DR

This work addresses the vulnerability of graph diffusion methods to adversarial graph structure changes and noisy node features. It introduces a min-max Adversarial Graph Signal Denoising (AGSD) formulation that yields Graph Adversarial Diffusion Convolution (GADC), incorporating a robustness term via a perturbed Laplacian and a diffusion-based approximation for scalability. The main contributions are (i) a closed-form inner solution for the adversarial perturbation, (ii) a scalable outer diffusion that includes an additional term to adapt to edge perturbations and feature noise, and (iii) demonstrated improvements on heterophilic graphs and across diverse datasets, with extensive ablations and defenses against adaptive attacks. The practical impact lies in providing a robust, diffusion-based GNN variant that maintains performance under adversarial structure perturbations and noisy features, while also enabling efficient precomputation and deployment for large graphs.

Abstract

This paper introduces a min-max optimization formulation for the Graph Signal Denoising (GSD) problem. In this formulation, we first maximize the second term of GSD by introducing perturbations to the graph structure based on Laplacian distance and then minimize the overall loss of the GSD. By solving the min-max optimization problem, we derive a new variant of the Graph Diffusion Convolution (GDC) architecture, called Graph Adversarial Diffusion Convolution (GADC). GADC differs from GDC by incorporating an additional term that enhances robustness against adversarial attacks on the graph structure and noise in node features. Moreover, GADC improves the performance of GDC on heterophilic graphs. Extensive experiments demonstrate the effectiveness of GADC across various datasets. Code is available at https://github.com/SongtaoLiu0823/GADC.

Graph Adversarial Diffusion Convolution

TL;DR

This work addresses the vulnerability of graph diffusion methods to adversarial graph structure changes and noisy node features. It introduces a min-max Adversarial Graph Signal Denoising (AGSD) formulation that yields Graph Adversarial Diffusion Convolution (GADC), incorporating a robustness term via a perturbed Laplacian and a diffusion-based approximation for scalability. The main contributions are (i) a closed-form inner solution for the adversarial perturbation, (ii) a scalable outer diffusion that includes an additional term to adapt to edge perturbations and feature noise, and (iii) demonstrated improvements on heterophilic graphs and across diverse datasets, with extensive ablations and defenses against adaptive attacks. The practical impact lies in providing a robust, diffusion-based GNN variant that maintains performance under adversarial structure perturbations and noisy features, while also enabling efficient precomputation and deployment for large graphs.

Abstract

This paper introduces a min-max optimization formulation for the Graph Signal Denoising (GSD) problem. In this formulation, we first maximize the second term of GSD by introducing perturbations to the graph structure based on Laplacian distance and then minimize the overall loss of the GSD. By solving the min-max optimization problem, we derive a new variant of the Graph Diffusion Convolution (GDC) architecture, called Graph Adversarial Diffusion Convolution (GADC). GADC differs from GDC by incorporating an additional term that enhances robustness against adversarial attacks on the graph structure and noise in node features. Moreover, GADC improves the performance of GDC on heterophilic graphs. Extensive experiments demonstrate the effectiveness of GADC across various datasets. Code is available at https://github.com/SongtaoLiu0823/GADC.
Paper Structure (54 sections, 4 theorems, 32 equations, 2 figures, 14 tables, 1 algorithm)

This paper contains 54 sections, 4 theorems, 32 equations, 2 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations.
  4. Spectral Graph Convolution.
  5. Graph Convolution Network (GCN).
  6. Adversarial Training.
  7. Graph Diffusion Convolution (GDC).
  8. Graph Adversarial Diffusion Convolution
  9. Adversarial Graph Signal Denoising Problem
  10. Problem Formulation
  11. Laplacian Distance.
  12. AGSD.
  13. Closed-form Solution of the Inner Maximization Problem
  14. Modified Transition Matrix Derived by Outer Minimization Problem
  15. Approximate Solution of Eq. \ref{['eq:graph adversarial diffusion inverse']}.
  16. ...and 39 more sections

Key Result

Proposition 1

Each entry of the noise matrix $\bm{\Upsilon}$, i.e., $[\bm{\Upsilon}]_{ij}$ is i.i.d sub-Gaussian random variable with variance $\sigma$ and mean $\mu=0$, i.e.,

Figures (2)

  • Figure 1: Illustration of the inner maximization problem. The loss function reaches the largest value when the direction of $\bm{\delta}^{\top}$ is the same as $\nabla h(\bm{\delta})$.
  • Figure 2: An illustration of $\tau$ on various graph structures. $\mathcal{G}_1$: nodes are isolated; $\mathcal{G}_2$: a star graph with 4 nodes; $\mathcal{G}_3$: a complete graph with 4 nodes. For computing $\tau$, we set $\lambda$ and $K$ as 32. $\tau$ has a smaller value if the graph has good connectivity.

Theorems & Definitions (7)

  • Proposition 1: Noise Property
  • Remark 1
  • Theorem 1: Upper Bound
  • Lemma 1
  • proof
  • Lemma 2
  • proof : Proof of Theorem 1.