  Implicit Graph Neural Networks (GNNs) have achieved significant success in
addressing graph learning problems recently. However, poorly designed implicit
GNN layers may have limited adaptability to learn graph metrics, experience
over-smoothing issues, or exhibit suboptimal convergence and generalization
properties, potentially hindering their practical performance. To tackle these
issues, we introduce a geometric framework for designing implicit graph
diffusion layers based on a parameterized graph Laplacian operator. Our
framework allows learning the metrics of vertex and edge spaces, as well as the
graph diffusion strength from data. We show how implicit GNN layers can be
viewed as the fixed-point equation of a Dirichlet energy minimization problem
and give conditions under which it may suffer from over-smoothing during
training (OST) and inference (OSI). We further propose a new implicit GNN model
to avoid OST and OSI. We establish that with an appropriately chosen
hyperparameter greater than the largest eigenvalue of the parameterized graph
Laplacian, DIGNN guarantees a unique equilibrium, quick convergence, and strong
generalization bounds. Our models demonstrate better performance than most
implicit and explicit GNN baselines on benchmark datasets for both node and
graph classification tasks.
