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Resilient Graph Neural Networks: A Coupled Dynamical Systems Approach

Moshe Eliasof, Davide Murari, Ferdia Sherry, Carola-Bibiane Schönlieb

TL;DR

This paper introduces an innovative approach to fortify GNNs against adversarial perturbations through the lens of coupled dynamical systems, which introduces graph neural layers based on differential equations with contractive properties, which improve the robustness of GNNs.

Abstract

Graph Neural Networks (GNNs) have established themselves as a key component in addressing diverse graph-based tasks. Despite their notable successes, GNNs remain susceptible to input perturbations in the form of adversarial attacks. This paper introduces an innovative approach to fortify GNNs against adversarial perturbations through the lens of coupled dynamical systems. Our method introduces graph neural layers based on differential equations with contractive properties, which, as we show, improve the robustness of GNNs. A distinctive feature of the proposed approach is the simultaneous learned evolution of both the node features and the adjacency matrix, yielding an intrinsic enhancement of model robustness to perturbations in the input features and the connectivity of the graph. We mathematically derive the underpinnings of our novel architecture and provide theoretical insights to reason about its expected behavior. We demonstrate the efficacy of our method through numerous real-world benchmarks, reading on par or improved performance compared to existing methods.

Resilient Graph Neural Networks: A Coupled Dynamical Systems Approach

TL;DR

This paper introduces an innovative approach to fortify GNNs against adversarial perturbations through the lens of coupled dynamical systems, which introduces graph neural layers based on differential equations with contractive properties, which improve the robustness of GNNs.

Abstract

Graph Neural Networks (GNNs) have established themselves as a key component in addressing diverse graph-based tasks. Despite their notable successes, GNNs remain susceptible to input perturbations in the form of adversarial attacks. This paper introduces an innovative approach to fortify GNNs against adversarial perturbations through the lens of coupled dynamical systems. Our method introduces graph neural layers based on differential equations with contractive properties, which, as we show, improve the robustness of GNNs. A distinctive feature of the proposed approach is the simultaneous learned evolution of both the node features and the adjacency matrix, yielding an intrinsic enhancement of model robustness to perturbations in the input features and the connectivity of the graph. We mathematically derive the underpinnings of our novel architecture and provide theoretical insights to reason about its expected behavior. We demonstrate the efficacy of our method through numerous real-world benchmarks, reading on par or improved performance compared to existing methods.
Paper Structure (39 sections, 6 theorems, 87 equations, 11 figures, 16 tables)

This paper contains 39 sections, 6 theorems, 87 equations, 11 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Outline of the paper.
  4. Related Work
  5. Graph Neural Networks as Dynamical Systems.
  6. Adversarial Defense in Graph Neural Networks.
  7. Preliminaries
  8. Measuring graph attacks.
  9. Method
  10. Graph Neural Networks Inspired by Coupled Contractive Systems
  11. Contractive Node Feature Dynamical System
  12. Contractive Adjacency Matrix Dynamical System
  13. Experiments
  14. Experimental settings
  15. Datasets.
  16. ...and 24 more sections

Key Result

Theorem 4.1

Assume $\sigma$ is a monotonically increasing 1-Lipschitz non-linear function. There are choices of $({\bf W}_l,{\bf K}_l)\in\mathbb{R}^{c\times c}\times \mathbb{R}^{c\times c}$, for which the explicit Euler step in eq:nodeFeatureDynamics is stable for a small enough $h_l>0$, i.e. there is a convex

Figures (11)

  • Figure 1: The coupled dynamical system $\mathcal{D}$ in CSGNN, as formulated in \ref{['eq:D']}.
  • Figure 2: Node classification accuracy (%) under nettack. The horizontal axis describes the number of perturbations per node.
  • Figure 3: Node classification accuracy (%) under a random adjacency matrix attack. The horizontal axis describes the attack percentage.
  • Figure 4: Node classification accuracy (%) using unit-tests from mujkanovicAreDefensesGraph2022. Results are relative to a baseline GCN. The horizontal axis shows the attack budget (%).
  • Figure 5: Node classification accuracy (%) under targeted attack with nettack to both node features and adjacency matrix. The horizontal axis describes the number of perturbations per node.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 3.1: $(\varepsilon_1,\varepsilon_2)-$robust GNN
  • Theorem 4.1: \ref{['eq:nodeFeatureDynamics']} can induce stable node dynamics
  • Theorem 4.2: \ref{['eq:nodeFeatureDynamics']} induces contractive node dynamics
  • Theorem 4.3: \ref{['eq:isContractive']} defines contractive adjacency dynamics
  • Definition A.1: Contractive dynamical system
  • Definition D.1: Graph gradient operator
  • Theorem E.1
  • proof
  • Theorem E.2
  • proof
  • ...and 2 more