Homomorphism Counts for Graph Neural Networks: All About That Basis

TL;DR

This paper explores the expressive power of graph neural networks, proposing a basis-based approach that counts all structures in the pattern basis, leading to more powerful models without extra computational cost.

Contribution

It introduces a novel basis-based method for counting homomorphisms, enhancing GNN expressiveness beyond existing pattern counting approaches.

Findings

Basis-based counts are strictly more expressive.

The approach matches the computational complexity of existing methods.

Empirical validation confirms theoretical advantages.

Abstract

A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at the heart of such limitations, since many functions to be learned rely on the ability of counting such patterns. Two prominent paradigms aim to address this limitation by enriching the graph features with subgraph or homomorphism pattern counts. In this work, we show that both of these approaches are sub-optimal in a certain sense and argue for a more fine-grained approach, which incorporates the homomorphism counts of all structures in the ``basis'' of the target pattern. This yields strictly more expressive architectures without incurring any additional overhead in terms of computational complexity compared to existing approaches. We prove a series of theoretical results on node-level and graph-level motif parameters and empirically validate them on standard benchmark datasets.