Lyapunov Stable Graph Neural Flow

TL;DR

This paper introduces a control theory-based framework for enhancing GNN robustness using Lyapunov stability, providing theoretical guarantees and improved performance against adversarial attacks.

Contribution

It presents a novel Lyapunov stability-based defense mechanism for GNNs, integrating adaptive, learnable Lyapunov functions with a projection mechanism for robustness.

Findings

Significantly improves robustness against adversarial attacks.

Achieves theoretical stability guarantees for GNNs.

Outperforms existing methods on standard benchmarks.

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.