Homomorphism Counts as Structural Encodings for Graph Learning

TL;DR

This paper introduces motif structural encoding (MoSE), a novel graph structural encoding based on homomorphism counts, which enhances graph neural network expressiveness and achieves state-of-the-art results in molecular property prediction.

Contribution

The paper proposes MoSE, a new structural encoding framework based on homomorphism counts, and demonstrates its superior expressiveness and empirical performance over existing encodings.

Findings

MoSE outperforms existing encodings across various architectures.

MoSE achieves state-of-the-art results on molecular property prediction.

Theoretically, MoSE's expressive power surpasses or matches other encodings.

Abstract

Graph Transformers are popular neural networks that extend the well-known Transformer architecture to the graph domain. These architectures operate by applying self-attention on graph nodes and incorporating graph structure through the use of positional encodings (e.g., Laplacian positional encoding) or structural encodings (e.g., random-walk structural encoding). The quality of such encodings is critical, since they provide the necessary $\textit{graph inductive biases}$ to condition the model on graph structure. In this work, we propose $\textit{motif structural encoding}$ (MoSE) as a flexible and powerful structural encoding framework based on counting graph homomorphisms. Theoretically, we compare the expressive power of MoSE to random-walk structural encoding and relate both encodings to the expressive power of standard message passing neural networks. Empirically, we observe that MoSE outperforms other well-known positional and structural encodings across a range of architectures, and it achieves state-of-the-art performance on a widely studied molecular property prediction dataset.