  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.
