Graph Transformers Dream of Electric Flow

Xiang Cheng, Lawrence Carin, Suvrit Sra

Keywords: Transformer, Graph Neural Network

TL;DR: we provide simple constructions showing how linear Transformers can learn to solve a number of canonical graph problems, with practical implications for learning positional encodings.

Abstract:

We show theoretically and empirically that the linear Transformer, when applied to graph data, can implement algorithms that solve canonical problems such as electric flow and eigenvector decomposition. The Transformer has access to information on the input graph only via the graph's incidence matrix. We present explicit weight configurations for implementing each algorithm, and we bound the constructed Transformers' errors by the errors of the underlying algorithms. Our theoretical findings are corroborated by experiments on synthetic data. Additionally, on a real-world molecular regression task, we observe that the linear Transformer is capable of learning a more effective positional encoding than the default one based on Laplacian eigenvectors. Our work is an initial step towards elucidating the inner-workings of the Transformer for graph data. Code is available at https://github.com/chengxiang/LinearGraphTransformer

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 7996