[论文解读] On the Expressive Power of Geometric Graph Neural Networks

Chaitanya K. Joshi, Cristian Bodnar+3 more|arXiv (Cornell University)|Jan 23, 2023

Computational Drug Discovery Methods被引用 24

一句话总结

论文介绍 Geometric Weisfeiler-Leman (GWL) test 以表征在物理对称性下几何 GNN 的表达能力，并分析 invariant/equivariant 层、higher-order tensors、以及 scalarisation 如何影响几何图的判别。

ABSTRACT

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at \url{https://github.com/chaitjo/geometric-gnn-dojo}

研究动机与目标

• Motivate the need to study expressive power for geometric graphs embedded in Euclidean space under physical symmetries.
• Develop a geometric variant of the Weisfeiler-Leman test (GWL) to distinguish geometric graphs while respecting permutations, rotations, reflections, and translations.
• Characterize how design choices in geometric GNNs (invariant vs equivariant layers, higher-order tensors, scalarisation) influence expressivity.
• Demonstrate equivalence between GWL discrimination and universal approximation of geometric GNNs.
• Provide synthetic experiments supporting the theoretical findings.

提出的方法

• Define GWL to discriminate geometric graphs while enforcing permutation, rotation, reflection, and translation symmetries.
• Analyze invariant layers and show their limited ability to distinguish one-hop identical geometric graphs.
• Show that equivariant layers propagate geometric information beyond local neighborhoods and increase expressivity.
• Incorporate higher-order tensors and scalarisation to achieve maximal expressivity.
• Prove that the GWL discrimination perspective is equivalent to universal approximation for geometric GNNs.

实验结果

研究问题

• RQ1 Can GNNs that are invariant to physical symmetries distinguish geometric graphs as effectively as a symmetry-respecting WL test?
• RQ2 How do invariant versus equivariant layers compare in their ability to distinguish geometric graphs?
• RQ3 What role do higher-order tensors and scalarisation play in maximizing the expressivity of geometric GNNs?
• RQ4 Is the GWL-based discrimination perspective equivalent to universal approximation for geometric GNNs?

主要发现

• Invariant layers have limited expressivity and cannot distinguish certain one-hop identical geometric graphs.
• Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighborhoods.
• Higher-order tensors and scalarisation enable geometrically expressive GNNs with maximal power.
• GWL's discrimination-based view aligns with universal approximation for geometric GNNs.
• Synthetic experiments supplement the theoretical results.