Capturing Graphs with Hypo-Elliptic Diffusions

TL;DR

This paper introduces the hypo-elliptic graph Laplacian, a new tensor-valued operator for graph neural networks that captures long-range dependencies efficiently and robustly, outperforming traditional methods in long-range reasoning tasks.

Contribution

It extends graph diffusion models using hypo-elliptic diffusions, providing a novel operator with theoretical guarantees and scalable algorithms for long-range graph reasoning.

Findings

Competitive with graph transformers on long-range reasoning datasets

Scales linearly with the number of edges, not quadratically with nodes

Provides theoretical guarantees and efficient low-rank approximations

Abstract

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.