[論文レビュー] Lyapunov Stable Graph Neural Flow

Haoyu Chu, Xiaotong Chen+5 more|arXiv (Cornell University)|Mar 13, 2026

Advanced Graph Neural Networks被引用数 0

ひとこと要約

We integrate Lyapunov stability theory with graph neural flows (integer- and fractional-order) to achieve provable robustness against adversarial perturbations, via a learnable Lyapunov function and a projection mechanism. The approach is orthogonal to existing defenses and can be combined with adversarial training.

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.

研究の動機と目的

• Motivate robustness for GNNs against topology and feature perturbations.
• Bridge graph neural flows with Lyapunov stability to provide theoretical guarantees.
• Propose a learnable Lyapunov function and projection to enforce stability.
• Enable compatibility with existing defenses, including adversarial training.
• Demonstrate robustness improvements across multiple datasets and attack types.

提案手法

• Model node feature updates as integer- or fractional-order dynamical systems (ODEs/FDEs).
• Introduce a Lyapunov stability module with a learnable Lyapunov function V implemented via an ICNN to enforce stability.
• Project base dynamics onto a Lyapunov-stable space using a defined projection operation.
• Include an equilibrium-separating classification layer to maximize inter-class separation while preserving stability.
• Provide theoretical guarantees: exponential stability for integer-order and Mittag-Leffler stability for fractional-order systems.
• Support multiple base dynamical systems (GRAND, GraphBel, GraphCON and their fractional variants) as the underlying dynamics.

実験結果

リサーチクエスチョン

• RQ1 Can Lyapunov stability theory provide rigorous, certificate-based robustness guarantees for graph neural flows (both integer- and fractional-order)?
• RQ2 How can we construct a learnable Lyapunov function that ensures stability without sacrificing predictive performance?
• RQ3 Does projecting base GNN dynamics into a Lyapunov-stable space improve robustness under various adversarial attacks (white-box, black-box, poisoning, evasion)?
• RQ4 Can the stability module be integrated with existing defenses to achieve cumulative robustness without excessive computation?
• RQ5 Does an equilibrium-separating layer enhance class separation while maintaining stability guarantees?

主な発見

Dataset	Attack	GRAND	IL-GRAND	F-GRAND	FL-GRAND	GBel	IL-GBel	F-GBel	FL-GBel	GCON	IL-GCON	F-GCON	FL-GCON	HANG
Cora	clean	82.24±1.82	88.88±0.97	86.44±0.31	88.95±0.39	79.07±0.46	85.73±0.73	77.55±0.79	86.74±0.93	83.10±0.63	86.97±0.60	82.42±0.89	87.10±0.80	87.13±0.86
Cora	PGD	36.80±1.86	79.01±1.24	56.38±6.39	80.76±0.33	63.93±3.88	80.86±0.73	69.50±2.83	82.60±0.67	48.38±2.44	72.71±0.38	56.70±4.36	74.28±0.69	78.37±1.84
Citeseer	clean	72.52±0.73	75.47±0.64	71.91±0.43	75.66±0.48	74.75±0.28	74.58±0.70	71.09±0.30	75.03±0.47	72.07±0.93	63.08±1.02	73.50±0.43	74.61±0.80	74.11±0.62
Pubmed	clean	88.44±0.34	86.89±0.65	88.39±0.47	87.43±0.42	88.18±1.89	90.25±0.20	89.51±0.12	93.78±0.13	88.09±0.32	86.73±0.25	90.30±0.11	87.27±0.74	89.93±0.27

• The proposed IL-GNNs and FL-GNNs achieve robustness by projecting dynamics onto a Lyapunov-stable space, with theoretical exponential (integer-order) or Mittag-Leffler (fractional-order) stability.
• Empirical results show substantial robustness gains across standard benchmarks and diverse attack scenarios, outperforming base models and several baselines.
• The stability module uses a learnable Lyapunov function V built from an ICNN to ensure V is positive definite and decreases along trajectories.
• The approach is compatible with adversarial training, enabling cumulative robustness when combined with existing defenses.
• An equilibrium-separating layer helps maximize inter-class distance between equilibria while preserving stability.