k-Maximum Inner Product Attention for Graph Transformers and the Expressive Power of GraphGPS

TL;DR

This paper introduces k-MIP attention for graph transformers, achieving linear memory complexity and high scalability while maintaining expressive power, enabling processing of large graphs efficiently.

Contribution

The paper proposes k-MIP attention, a sparse, top-k based attention mechanism that balances efficiency and expressive power in large-scale graph transformers.

Findings

k-MIP attention achieves linear memory complexity.

Enables processing graphs with over 500k nodes on a single GPU.

Maintains full expressive power comparable to full-attention transformers.

Abstract

Graph transformers have shown promise in overcoming limitations of traditional graph neural networks, such as oversquashing and difficulties in modeling long-range dependencies. However, their application to large-scale graphs is hindered by the quadratic memory and computational complexity of the all-to-all attention mechanism. Although alternatives such as linearized attention and restricted attention patterns have been proposed, these often degrade performance or limit expressive power. To better balance efficiency and effectiveness, we introduce k-Maximum Inner Product (k-MIP) attention for graph transformers. k-MIP attention selects the most relevant key nodes per query via a top-k operation, yielding a sparse yet flexible attention pattern. Combined with an attention score computation based on symbolic matrices, this results in linear memory complexity and practical speedups of up to an order of magnitude compared to all-to-all attention, enabling the processing of graphs with over 500k nodes on a single A100 GPU. We provide a theoretical analysis of expressive power, showing that k-MIP attention does not compromise the expressiveness of graph transformers: specifically, we prove that k-MIP transformers can approximate any full-attention transformer to arbitrary precision. In addition, we analyze the expressive power of the GraphGPS framework, in which we integrate our attention mechanism, and establish an upper bound on its graph distinguishing capability in terms of the S-SEG-WL test. Finally, we validate our approach on the Long Range Graph Benchmark, the City-Networks benchmark, and two custom large-scale inductive point cloud datasets, consistently ranking among the top-performing scalable graph transformers.