Overview of CT-Layer: A Differentiable and Learnable Rewiring Layer
CT-Layer is a graph neural network layer that is able to rewire a graph in an inductive and parameter-free way according to the commute times distance or effective resistance. CT-Layer addresses the issue of learning a differentiable way to compute the CT-embedding of the graph, which is not possible with the traditional spectral version. CT-Layer provides a new approach to rewire a given graph optimally, leading to a better understanding of the graph structure.
Problem with Spectral CT-Embedding
CT-embedding, which is used as a measure of the distance between nodes, is usually computed spectrally in the literature. This non-differentiable computation process poses as an issue for proposing differentiable methods to learn the embeddings. Additionally, CT-embedding is approximated using the heat kernel, which is dependent on the hyperparameter.
CT-Layer as an Optimization Problem
CT-Layer formulates the problem giving that CT-embedding minimizes Dirichlet energies given by the constraint of neighboring nodes to have similar embeddings, resulting in an orthogonal and normalized matrix. CT-Layer is then defined as the rewiring layer given the matrix encoding the features of the nodes, which learns the association and optimizes according to the loss. This results in the resistance diffusion, which provides input to the subsequent MP layer.
Preservation of Structure
CT-Layer rewires the graph based on the commute times distance, but does it preserve the original structure? The preservation of the original structure is important in any graph neural network model. The sampling algorithm of graph G handles this issue by sampling edges from G with probabilities proportional to the effective resistance. The Dirichlet energies in G are bounded in (1±ϵ) of the Dirichlet energies of the original graph, ensuring the preservation of the original structure.
Conclusion
CT-Layer offers a new learnable and differentiable way to calculate the commute times embedding and distance of the graph. CT-Layer allows for the optimization of the embeddings in a differentiable way, leading to a better understanding of the graph structure. CT-Layer's approach ensures the preservation of the original structure, which is essential for any graph neural network model.
