Persistent Gaussian Perturbations Prevent Oversmoothing in Recurrent Graph Neural Networks

Graph neural networks (GNNs) often suffer from oversmoothing, where repeated message passing causes node representations to converge to a constant vector, losing all discriminative power. This paper investigates whether persistent stochastic perturbations—specifically, injecting Gaussian noise after each propagation step—can fundamentally alter the long-term dynamics of recurrent GNNs to prevent this collapse.

The author models the noisy recurrent GNN as a stochastic dynamical system. By treating the hidden representations as a Markov chain, the paper applies tools from spectral graph theory and ergodic theory to analyze the system's long-term behavior. The analysis focuses on the Dirichlet energy, a measure of representation smoothness, and derives a lower bound for its stationary value based on the noise variance and the spectral gap of the graph Laplacian.

The study establishes that under a global contraction assumption, the noisy recurrent GNN converges to a unique invariant probability measure. Crucially, the expected stationary Dirichlet energy is bounded below by a term proportional to the noise variance and the graph's spectral gap. This provides a rigorous guarantee that the representations cannot collapse onto the constant manifold, effectively preventing asymptotic oversmoothing. Numerical experiments on both linear and nonlinear recurrent architectures confirm that the stationary Dirichlet energy scales predictably with the noise intensity.

This work shifts the view of noise in GNNs from a mere regularization heuristic to a fundamental mechanism for controlling representation geometry. By providing an explicit link between noise levels, graph topology, and representation diversity, the paper offers a principled framework for designing deep or equilibrium-based GNNs that maintain expressive power over arbitrary depths.