On the Stability of Expressive Positional Encodings for Graphs (2310.02579v3)

Abstract: Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks. Although widespread, using Laplacian eigenvectors as positional encodings faces two fundamental challenges: (1) \emph{Non-uniqueness}: there are many different eigendecompositions of the same Laplacian, and (2) \emph{Instability}: small perturbations to the Laplacian could result in completely different eigenspaces, leading to unpredictable changes in positional encoding. Despite many attempts to address non-uniqueness, most methods overlook stability, leading to poor generalization on unseen graph structures. We identify the cause of instability to be a hard partition'' of eigenspaces. Hence, we introduce Stable and Expressive Positional Encodings (SPE), an architecture for processing eigenvectors that uses eigenvalues tosoftly partition'' eigenspaces. SPE is the first architecture that is (1) provably stable, and (2) universally expressive for basis invariant functions whilst respecting all symmetries of eigenvectors. Besides guaranteed stability, we prove that SPE is at least as expressive as existing methods, and highly capable of counting graph structures. Finally, we evaluate the effectiveness of our method on molecular property prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods. Our code is available at \url{https://github.com/Graph-COM/SPE}.

On the Stability of Expressive Positional Encodings for Graphs

The effectiveness of graph transformers and message-passing graph neural networks (GNNs) hinges significantly on their ability to incorporate positional encodings, impacting their performance on tasks across domains such as drug discovery and social network analysis. This paper tackles two critical issues associated with the prevalent use of Laplacian eigenvectors in graph positional encodings: non-uniqueness and instability. Recognizing that most attempts have inadequately addressed stability, this work introduces a novel approach termed Stable and Expressive Positional Encodings (SPE), demonstrating both theoretical soundness and empirical effectiveness.

Key Contributions

1. Introduction of SPE: The authors propose SPE, which leverages a soft partitioning strategy based on eigenvalues, ensuring both stability and expressivity. This marks a significant advancement over conventional methods that typically employ hard partitions of eigenspaces leading to sensitivity to perturbations. SPE uniquely balances stability through continuous soft partitioning with expressivity derived from eigenvalue-dependent processing.
2. Theoretical Guarantees: SPE is analytically shown to be stable; the network’s sensitivity to input perturbations scales with the eigengap between the $d$-th and $(d+1)$-th eigenvalues. This provides a solid footing for the model to generalize across unseen graph structures, a crucial attribute in dynamic real-world applications.
3. Empirical Validation: Extensive experiments on molecular property prediction and OOD generalization tasks exhibit superior generalization capabilities of SPE, outperforming prior methods in scenarios with domain shifts, as evidenced on benchmarks like ZINC and Alchemy.
4. Expressivity and Universality: The paper theoretically verifies SPE’s capability to approximate any continuous basis invariant function, equating its expressivity to that of BasisNet while surpassing it in practical robustness due to its stability.
5. Practical Implications: The empirical results, particularly those involving OOD generalization tasks, underscore SPE's potential in applications that require robustness to domain shifts, such as bioinformatics and social network analysis. The stability ensures that small perturbations in input graphs do not result in erratic changes in the positional encodings, which is essential for reliable deployment in critical applications.
6. Trade-off Analysis: The discussion about the trade-off between stability and expressivity provides deep insights into the nuanced design decisions required to optimize GNNs for specific tasks, presenting SPE as a versatile tool that can be adjusted based on the application’s tolerance for generalization error versus the need for detailed expressivity.

Future Directions

The work opens several avenues for future research:

• Extension to Larger and More Complex Graphs:

Investigating the scalability of SPE to very large graph datasets, which frequently appear in industrial applications, could further solidify its applicability.

• Application to Diverse Graph-based Tasks:

Evaluating SPE on tasks such as link prediction or community detection in dynamic and heterogeneous graphs could demonstrate broader utility.

• Integration with Novel GNN Architectures:

Exploring the integration of SPE with emerging GNN architectures, perhaps inspired by advances in LLMs or other areas of machine learning, could yield enhanced performance by combining strengths from different paradigms.

In sum, the paper makes a notable contribution to the field of GNNs by resolving a significant hindrance in positional encodings, presenting a methodology that is both theoretically robust and empirically superior in enhancing the adaptability and performance of graph-based learning models.