Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks

TL;DR

This paper introduces PEG, a new class of GNN layers that incorporate equivariant positional encoding, improving generalization and scalability in graph tasks like link prediction.

Contribution

The paper proposes PEG, a mathematically grounded GNN layer that uses orthogonal group equivariance for positional features, enhancing stability and inductive generalization.

Findings

PEG outperforms existing methods in link prediction accuracy.

PEG demonstrates better generalization to unseen graphs.

PEG scales efficiently to large real-world networks.

Abstract

Graph neural networks (GNN) have shown great advantages in many graph-based learning tasks but often fail to predict accurately for a task-based on sets of nodes such as link/motif prediction and so on. Many works have recently proposed to address this problem by using random node features or node distance features. However, they suffer from either slow convergence, inaccurate prediction, or high complexity. In this work, we revisit GNNs that allow using positional features of nodes given by positional encoding (PE) techniques such as Laplacian Eigenmap, Deepwalk, etc. GNNs with PE often get criticized because they are not generalizable to unseen graphs (inductive) or stable. Here, we study these issues in a principled way and propose a provable solution, a class of GNN layers termed PEG with rigorous mathematical analysis. PEG uses separate channels to update the original node features and positional features. PEG imposes permutation equivariance w.r.t. the original node features and imposes $O(p)$ (orthogonal group) equivariance w.r.t. the positional features simultaneously, where $p$ is the dimension of used positional features. Extensive link prediction experiments over 8 real-world networks demonstrate the advantages of PEG in generalization and scalability.