## Abstract

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.

Original language | English |
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Journal | Advances in Neural Information Processing Systems |

Volume | 36 |

State | Published - 2023 |

Event | 37th Conference on Neural Information Processing Systems, NeurIPS 2023 - New Orleans, United States Duration: 10 Dec 2023 → 16 Dec 2023 |

### Funding

Funders | Funder number |
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Blavatnik Family Foundation | |

Yandex Initiative in Machine Learning | |

Adelis Research Fund for Artificial Intelligence | |

Tel Aviv University | |

Amnon and Anat Shashua | |

Google Research Gift | |

Israel Science Foundation | 1780/21 |

Israel Science Foundation |