Stuart-Landau Oscillatory Graph Neural Network

TL;DR

The paper tackles oversmoothing and vanishing gradients in deep GNNs by introducing SLGNN, a complex-valued graph neural network based on coupled Stuart-Landau oscillators that evolve both amplitude $r$ and phase $φ$ of node features. It develops a bespoke implicit-explicit (IMEX) time-stepping scheme to stabilize training by treating the stiff $|z|^2z$ term implicitly while handling coupling explicitly, with a magnitude-phase decomposition facilitating efficient updates. The authors provide a thorough theoretical and phenomenological comparison among Stuart-Landau, Kuramoto, and harmonic oscillators, then demonstrate that SLGNN achieves state-of-the-art performance across node classification, graph classification, and graph regression benchmarks, particularly when leveraging deeper architectures. The work shows that amplitude dynamics yield richer expressiveness, near-critical operating regimes, and robust performance under perturbations, offering a practical and theoretically grounded framework for deep oscillatory graph architectures with significant potential impact on complex graph tasks $\left( r, φ, α, β, ω, γ \right)$.

Abstract

Oscillatory Graph Neural Networks (OGNNs) are an emerging class of physics-inspired architectures designed to mitigate oversmoothing and vanishing gradient problems in deep GNNs. In this work, we introduce the Complex-Valued Stuart-Landau Graph Neural Network (SLGNN), a novel architecture grounded in Stuart-Landau oscillator dynamics. Stuart-Landau oscillators are canonical models of limit-cycle behavior near Hopf bifurcations, which are fundamental to synchronization theory and are widely used in e.g. neuroscience for mesoscopic brain modeling. Unlike harmonic oscillators and phase-only Kuramoto models, Stuart-Landau oscillators retain both amplitude and phase dynamics, enabling rich phenomena such as amplitude regulation and multistable synchronization. The proposed SLGNN generalizes existing phase-centric Kuramoto-based OGNNs by allowing node feature amplitudes to evolve dynamically according to Stuart-Landau dynamics, with explicit tunable hyperparameters (such as the Hopf-parameter and the coupling strength) providing additional control over the interplay between feature amplitudes and network structure. We conduct extensive experiments across node classification, graph classification, and graph regression tasks, demonstrating that SLGNN outperforms existing OGNNs and establishes a novel, expressive, and theoretically grounded framework for deep oscillatory architectures on graphs.

Figures (3)

• Figure 1: Node classification accuracy (%) on Cora for GNNs of varying depths. Shaded regions indicate the 10th to 90th percentile range. The mean values are computed over 10 random splits and weight initializations.
• Figure 2: Robustness of GNNs on Cora dataset against random edge perturbations. Shaded regions indicate the 25th to 75th percentile range, computed over 10 random splits and weight initializations.
• Figure 3: Distribution of optimal layer depths selected by the hyperparameter tuner for different model types.