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On the Ability of Graph Neural Networks to Model Interactions Between Vertices

Noam RazinTom VerbinNadav Cohen
Nov 2022
摘要
Graph neural networks (GNNs) are widely used for modeling complexinteractions between entities represented as vertices of a graph. Despiterecent efforts to theoretically analyze the expressive power of GNNs, a formalcharacterization of their ability to model interactions is lacking. The currentpaper aims to address this gap. Formalizing strength of interactions through anestablished measure known as separation rank, we quantify the ability ofcertain GNNs to model interaction between a given subset of vertices and itscomplement, i.e. between sides of a given partition of input vertices. Ourresults reveal that the ability to model interaction is primarily determined bythe partition's walk index -- a graph-theoretical characteristic that we defineby the number of walks originating from the boundary of the partition.Experiments with common GNN architectures corroborate this finding. As apractical application of our theory, we design an edge sparsification algorithmnamed Walk Index Sparsification (WIS), which preserves the ability of a GNN tomodel interactions when input edges are removed. WIS is simple, computationallyefficient, and markedly outperforms alternative methods in terms of inducedprediction accuracy. More broadly, it showcases the potential of improving GNNsby theoretically analyzing the interactions they can model.
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