Graph neural networks (GNNs) are widely used for modeling complex interactions between entities represented as vertices of a graph. Despite recent efforts to theoretically analyze the expressive power of GNNs, a formal characterization of their ability to model interactions is lacking. The current paper aims to address this gap. Formalizing strength of interactions through an established measure known as separation rank, we quantify the ability of certain GNNs to model interaction between a given subset of vertices and its complement, i.e. between the sides of a given partition of input vertices. Our results reveal that the ability to model interaction is primarily determined by the partition's walk index -- a graph-theoretical characteristic defined by the number of walks originating from the boundary of the partition. Experiments with common GNN architectures corroborate this finding. As a practical application of our theory, we design an edge sparsification algorithm named Walk Index Sparsification (WIS), which preserves the ability of a GNN to model interactions when input edges are removed. WIS is simple, computationally efficient, and in our experiments has markedly outperformed alternative methods in terms of induced prediction accuracy. More broadly, it showcases the potential of improving GNNs by theoretically analyzing the interactions they can model.

翻译：图神经网络（GNN）广泛用于对以图中顶点表示的实体间的复杂交互进行建模。尽管近年来已有研究从理论上分析GNN的表达能力，但对其建模交互能力的正式刻画仍然缺失。本文旨在填补这一空白。通过一种称为分离秩（separation rank）的既定度量来形式化交互强度，我们量化了特定GNN在给定顶点子集与其补集之间（即给定输入顶点划分的双方之间）建模交互的能力。我们的结果表明，建模交互的能力主要由划分的游走指数（walk index）决定——这是一个由图划分边界出发的游走数量所定义的图论特征。对常见GNN架构的实验验证了这一发现。作为理论的实际应用，我们设计了一种名为游走指数稀疏化（Walk Index Sparsification, WIS）的边稀疏化算法，该算法能在移除输入边时保持GNN建模交互的能力。WIS方法简单、计算高效，且在实验中，其在诱导预测精度方面显著优于其他方法。更广泛而言，它展示了通过理论分析GNN所能建模的交互来改进GNN的潜力。