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）广泛用于对表示为图顶点实体间的复杂相互作用进行建模。尽管近期研究致力于从理论上分析GNN的表达能力，但目前仍缺乏对其建模相互作用能力的正式刻画。本文旨在填补这一空白。通过采用称为分离秩的成熟度量来形式化相互作用的强度，我们量化了特定GNN对给定顶点子集与其补集之间（即输入顶点划分两侧之间）相互作用建模的能力。研究结果表明，建模相互作用的能力主要取决于划分的游走指数——一种由划分边界出发的游走数量定义的图论特征。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（WIS）的边稀疏化算法，该算法能在删除输入边时保持GNN建模相互作用的能力。WIS算法简单、计算高效，并且在实验中，其在预测精度提升方面显著优于其他替代方法。更广泛而言，该工作展示了通过理论分析GNN可建模的相互作用来改进GNN的潜力。