  Implicit graph neural networks (GNNs) have emerged as a potential approach to
enable GNNs to capture long-range dependencies effectively. However, poorly
designed implicit GNN layers can experience over-smoothing or may have limited
adaptability to learn data geometry, potentially hindering their performance in
graph learning problems. To address these issues, we introduce a geometric
framework to design implicit graph diffusion layers based on a parameterized
graph Laplacian operator. Our framework allows learning the geometry of vertex
and edge spaces, as well as the graph gradient operator from data. We further
show how implicit GNN layers can be viewed as the fixed-point solution of a
Dirichlet energy minimization problem and give conditions under which it may
suffer from over-smoothing. To overcome the over-smoothing problem, we design
our implicit graph diffusion layer as the solution of a Dirichlet energy
minimization problem with constraints on vertex features, enabling it to trade
off smoothing with the preservation of node feature information. With an
appropriate hyperparameter set to be larger than the largest eigenvalue of the
parameterized graph Laplacian, our framework guarantees a unique equilibrium
and quick convergence. Our models demonstrate better performance than leading
implicit and explicit GNNs on benchmark datasets for node and graph
classification tasks, with substantial accuracy improvements observed for some
datasets.

Machine Learning