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 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 that we define 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 markedly outperforms 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:尽管最近努力从理论上分析GNNs的表达力,但目前缺乏关于其模拟互动能力的正式特征。本文旨在缩小这一差距。通过一种称为分离等级的既定措施,将互动强度正规化,我们量化某些GNNs在某一子节点的顶点及其补充之间(即输入顶点分配的两侧之间)建模互动的能力。我们的结果表明,建模互动的能力主要取决于分区的步行指数 -- -- 一种我们根据分区边界的行走次数界定的图表理论特征。与通用GNN结构的实验证实了这一发现。作为我们理论的实际应用,我们设计了一个称为行走指数Sparsification(WIS)的边边边边线调算算算法,这保留了GNN在删除输入边缘时建模互动的能力。WIS非常简单,具有计算效率,并且明显超出我们根据分区边界界限所定义的行走路线数目所定义的图理学特征特征特征。与通用GNNS结构的实验证实了这一发现。作为我们理论的实际应用,我们设计了一个称为行走指数Spariz(WS)的边调算算算算法,这保留了GNNNN的模型在模拟中可以广泛地改进各种方法。