Equivariant and Stable Positional Encoding for More Powerful Graph Neural 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 rotation equivariance w.r.t. the positional features simultaneously. Extensive link prediction experiments over 8 real-world networks demonstrate the advantages of PEG in generalization and scalability. 1

The prevalence and characteristics of metabolic syndrome according to different definitions in China: a nationwide cross-sectional study, 2012–2015

“…Equipped with the positional encoding, the graph transformers can go beyond 1-WL [95,96] and further proved to be universal [96]. However, such positional encodings are not permutationequivariant thus recent works attempt to design equivariant Laplacian positional encoding [144,145].…”

Universal Local Attractors on Graphs

“…To this end, the generic message passing scheme has been shown to be universal over Turing computable functions [1], although this does not cover the case of non-computable functions or density over certain sets of non-computable functions. More recent breakthroughs have shown that the expressive power is not necessarily a property of architectures alone, because enriching the node feature space with positional encodings [21,21,22], like random node representations, can make MPNNs more powerful [23]. A result that we use in later experiments is that MPNNs enriched with non-trainable node representations can express any non-attributed graph functions while retaining equivariance in probability [24].…”

Universal Local Attractors on Graphs

“…To this end, the generic message-passing scheme has been shown to be universal over Turing computable functions [1], although this does not cover the case of non-computable functions or density over certain sets of non-computable functions. More recent breakthroughs have shown that the expressive power is not necessarily a property of architectures alone, because enriching the node feature space with positional encodings [21,22], like random node representations, can make MPNNs more expressive [23]. A result that we use in later experiments is that MPNNs enriched with non-trainable node representations can express any non-attributed graph functions while retaining equivariance in probability [24].…”