Capturing Graphs with Hypo-Elliptic Diffusions
This provides a more efficient alternative to graph transformers for long-range reasoning tasks on graphs, though it appears incremental as an extension of existing diffusion-based approaches.
The paper tackles the problem of capturing long-range dependencies in graph neural networks by introducing a novel tensor-valued graph operator called the hypo-elliptic graph Laplacian, which extends random walk encodings using hypo-elliptic diffusions. Experiments show this method competes with graph transformers on datasets requiring long-range reasoning while scaling linearly in edges instead of quadratically in nodes.
Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator, which we call the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In particular, this gives a structured approach to capture long-range dependencies on graphs that is robust to pooling. Besides the attractive theoretical properties, our experiments show that this method competes with graph transformers on datasets requiring long-range reasoning but scales only linearly in the number of edges as opposed to quadratically in nodes.