Oversmoothing in Graph Neural Networks (GNNs) refers to the phenomenon where increasing network depth leads to homogeneous node representations. While previous work has established that Graph Convolutional Networks (GCNs) exponentially lose expressive power, it remains controversial whether the graph attention mechanism can mitigate oversmoothing. In this work, we provide a definitive answer to this question through a rigorous mathematical analysis, by viewing attention-based GNNs as nonlinear time-varying dynamical systems and incorporating tools and techniques from the theory of products of inhomogeneous matrices and the joint spectral radius. We establish that, contrary to popular belief, the graph attention mechanism cannot prevent oversmoothing and loses expressive power exponentially. The proposed framework extends the existing results on oversmoothing for symmetric GCNs to a significantly broader class of GNN models, including random walk GCNs, Graph Attention Networks (GATs) and (graph) transformers. In particular, our analysis accounts for asymmetric, state-dependent and time-varying aggregation operators and a wide range of common nonlinear activation functions, such as ReLU, LeakyReLU, GELU and SiLU.

翻译：图神经网络（GNNs）中的过平滑现象是指随着网络深度增加，节点表示趋于同质化的现象。虽然先前研究已证实图卷积网络（GCNs）会指数级丧失表达能力，但图注意力机制能否缓解过平滑仍存在争议。本文通过严格的数学分析，将基于注意力的GNNs视为非线性时变动力系统，并引入非齐次矩阵乘积理论与联合谱半径的技术工具，对该问题给出了明确解答。我们证明，与普遍认知相反，图注意力机制无法阻止过平滑，且会指数级丧失表达能力。本文提出的理论框架将现有针对对称GCNs的过平滑研究成果拓展至更广泛的GNN模型类别，包括随机游走GCNs、图注意力网络（GATs）及（图）Transformer。特别值得注意的是，我们的分析涵盖了非对称、状态依赖及时变聚合算子，以及ReLU、LeakyReLU、GELU和SiLU等常见非线性激活函数。