发表机构
Amirkabir University of Technology (Tehran Polytechnic)（阿米尔卡比尔理工大学（德黑兰理工大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对循环图神经网络的过平滑问题，提出注入持续高斯噪声的方法，从理论上证明该方法可保留表示多样性，实验也验证了其有效性。
AI 中文摘要
过平滑是深度图神经网络（GNN）的一项基本局限：重复的消息传递会导致节点表示越来越相似，最终坍缩至低维子空间。该现象限制了消息传递架构的有效深度，促使人们寻找能保留表示多样性的机制。本文研究一种循环图神经网络，其在每次传播步骤后注入独立高斯噪声，并将所得架构分析为随机动力系统。在确定性更新满足标准全局收缩假设的条件下，我们证明隐藏表示构成几何遍历马尔可夫链，存在唯一的不变概率测度。核心理论结果给出期望稳态狄利克雷能量的显式正下界，其与噪声方差及底层图的谱间隙成正比。因此，稳态表示无法坍缩至常数流形，为渐近过平滑的预防提供了严格保证——该保证基于非零狄利克雷能量。我们的分析揭示，持续随机扰动是与残差连接、归一化、图重连等确定性方法互补的、用于对抗过平滑的全新机制。最后，线性与非线性循环图神经网络的数值实验与理论预测高度吻合，验证了稳态分布的出现及极限狄利克雷能量对噪声强度的预期依赖关系。
英文摘要
Oversmoothing is a fundamental limitation of deep graph neural networks (GNNs), where repeated message passing causes node representations to become increasingly similar, eventually collapsing toward a low-dimensional subspace. This phenomenon limits the effective depth of message-passing architectures and motivates the search for mechanisms that preserve representation diversity. In this paper, we study a recurrent graph neural network in which independent Gaussian noise is injected after every propagation step and analyze the resulting architecture as a stochastic dynamical system. Under a standard global contraction assumption on the deterministic update, we prove that the hidden representations form a geometrically ergodic Markov chain admitting a unique invariant probability measure. Our main theoretical result establishes an explicit positive lower bound on the expected stationary Dirichlet energy, proportional to both the noise variance and the spectral gap of the underlying graph. Consequently, the stationary representations cannot collapse onto the constant manifold, providing a rigorous guarantee that asymptotic oversmoothing is prevented in the sense of non-vanishing Dirichlet energy. Our analysis reveals persistent stochastic perturbations as a fundamentally different mechanism for combating oversmoothing, complementing existing deterministic approaches based on residual connections, normalization, and graph rewiring. Finally, numerical experiments on both linear and nonlinear recurrent graph neural networks closely match the theoretical predictions, illustrating the emergence of a stationary distribution and the predicted dependence of the limiting Dirichlet energy on the noise intensity.
