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Capturing Graphs with Hypo-Elliptic Diffusions

Csaba Toth · Darrick Lee · Celia Hacker · Harald Oberhauser

Hall J (level 1) #429

Keywords: [ Graph Diffusion ] [ Graph Tensor Networks ] [ Random Walks ] [ Graph classification ] [ Hypo-Elliptic Laplacian ]


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.

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