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Lyapunov Stable Graph Neural Flow

Haoyu Chu, Xiaotong Chen, Wei Zhou, Wenjun Cui, Kai Zhao, Shikui Wei, Qiyu Kang

Abstract

Graph Neural Networks (GNNs) are highly vulnerable to adversarial perturbations in both topology and features, making the learning of robust representations a critical challenge. In this work, we bridge GNNs with control theory to introduce a novel defense framework grounded in integer- and fractional-order Lyapunov stability. Unlike conventional strategies that rely on resource-heavy adversarial training or data purification, our approach fundamentally constrains the underlying feature-update dynamics of the GNN. We propose an adaptive, learnable Lyapunov function paired with a novel projection mechanism that maps the network's state into a stable space, thereby offering theoretically provable stability guarantees. Notably, this mechanism is orthogonal to existing defenses, allowing for seamless integration with techniques like adversarial training to achieve cumulative robustness. Extensive experiments demonstrate that our Lyapunov-stable graph neural flows substantially outperform base neural flows and state-of-the-art baselines across standard benchmarks and various adversarial attack scenarios.

Lyapunov Stable Graph Neural Flow

Abstract

Graph Neural Networks (GNNs) are highly vulnerable to adversarial perturbations in both topology and features, making the learning of robust representations a critical challenge. In this work, we bridge GNNs with control theory to introduce a novel defense framework grounded in integer- and fractional-order Lyapunov stability. Unlike conventional strategies that rely on resource-heavy adversarial training or data purification, our approach fundamentally constrains the underlying feature-update dynamics of the GNN. We propose an adaptive, learnable Lyapunov function paired with a novel projection mechanism that maps the network's state into a stable space, thereby offering theoretically provable stability guarantees. Notably, this mechanism is orthogonal to existing defenses, allowing for seamless integration with techniques like adversarial training to achieve cumulative robustness. Extensive experiments demonstrate that our Lyapunov-stable graph neural flows substantially outperform base neural flows and state-of-the-art baselines across standard benchmarks and various adversarial attack scenarios.
Paper Structure (29 sections, 4 theorems, 33 equations, 3 figures, 6 tables)

This paper contains 29 sections, 4 theorems, 33 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Adversarial Attacks and Defenses on Graph
  4. Application of Lyapunov theory in Deep Learning
  5. Notation and Preliminaries
  6. Methodology
  7. The Base Graph Neural Flow
  8. Integer- and Fractional-Order Lyapunov Stability Module
  9. Lyapunov Stable GNNs and their Training Process
  10. Choices of the base dynamical system
  11. Details of stability module
  12. The equilibrium-separating classification layer
  13. The training process
  14. Certified Robustness under Perturbation
  15. Experiments
  16. ...and 14 more sections

Key Result

Theorem 2

Given a fixed point $\boldsymbol{u}^{\star}$, suppose there exists a continuously differentiable function $V: [0, \infty) \times \mathcal{U} \rightarrow \mathds{R}$ defined for $t \ge 0$ in a neighborhood $\mathcal{U}$ of $\boldsymbol{u}^{\star}$, fulfilling the following conditions: then $V$ is called a Lyapunov function, and the equilibrium $\boldsymbol{u}^{\star}$ is asymptotically stable. Fur

Figures (3)

  • Figure 1: The scratch of the Lyapunov stability module. The black dashed box illustrates the dynamic evolution of the base graph neural flow over the time interval $[0, t_n]$, where the fractional case captures long-range memories by incorporating historical states, unlike the integer-order's behavior depends solely on the current state. The orange dashed box depicts the projected Lyapunov-stable system, which satisfies the conditions of the Lyapunov's direct method, where the Lyapunov function is implemented by an input-convex neural network, a fully connected neural network with special properties.
  • Figure 2: Training time (s/epoch) and inference time (s) on the Cora dataset with $t\in[0,10]$ and step size of 1.
  • Figure 3: Influence of $\beta$ on the robust test accuracy on Cora dataset.

Theorems & Definitions (11)

  • Definition 1: Lyapunov stability
  • Theorem 2: Lyapunov's direct method giesl2015review
  • Definition 3: Mittag–Leffler stability
  • Remark 1
  • Theorem 4: Fractional Lyapunov's direct method LI20101810
  • Remark 2
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 1 more