Graph neural networks are architectures for learning invariant functions over
graphs. 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.

图神经网络是一种用于学习图上不变函数的架构。本研究证明了现有方法中存在的不足，并提出了一种更精细化的方法，通过整合目标模式中所有结构的同构计数，实现了更加具有表达能力的架构，而不增加额外的计算复杂度。我们在标准基准数据集上进行了一系列理论和实证验证。

图神经网络的同态计数：基础方面的研究

Homomorphism Counts for Graph Neural Networks: All About That Basis

We investigate novel random graph embeddings that can be computed in expected
polynomial time and that are able to distinguish all non-isomorphic graphs in
expectation. Previous graph embeddings have limited expressiveness and either
cannot distinguish all graphs or cannot be computed efficiently for every
graph. To be able to approximate arbitrary functions on graphs, we are
interested in efficient alternatives that become arbitrarily expressive with
increasing resources. Our approach is based on Lov\'asz' characterisation of
graph isomorphism through an infinite dimensional vector of homomorphism
counts. Our empirical evaluation shows competitive results on several benchmark
graph learning tasks.

本文研究了新颖的随机图嵌入算法，能在预期的多项式时间内区分所有非同构图，并基于 Lovasz 的同态计数理论实现对任意图的函数逼近，结果在多个基准图学习任务上表现竞争力。