使用高阶图神经网络计数子结构：可能性与不可能性结果
Counting Substructures with Higher-Order Graph Neural Networks: Possibility and Impossibility Results
AI总结：
本文提出一种新的局部邻域递归池化技术，使GNN能计数大小为k的子图，克服低阶GNN局限，并利用稀疏性降低计算复杂度，同时给出信息论与时间复杂度的下界。
摘要：
虽然消息传递图神经网络（GNNs）已成为图学习领域日益流行的架构，但近期研究揭示了其在表达能力上的重要缺陷。为此，研究者提出了若干高阶图神经网络，这些网络显著提升了表达能力，但代价是高昂的计算成本。受这一差距的驱动，我们探索了替代策略与下界。具体而言，我们分析了一种新的局部邻域递归池化技术，该技术允许在计算成本与表达能力之间进行不同的权衡。首先，我们证明该模型能够计数大小为$k$的子图，从而克服了低阶GNN的一个已知局限。其次，我们展示了递归池化如何利用稀疏性来降低计算复杂度，相较于现有高阶GNN更具优势。更一般地，我们为使用图表示（这些表示对派生（子）图的表示进行池化）来计数子图提供了（近乎）匹配的信息论下界。我们还讨论了时间复杂度方面的下界。
英文摘要：
While message passing Graph Neural Networks (GNNs) have become increasingly popular architectures for learning with graphs, recent works have revealed important shortcomings in their expressive power. In response, several higher-order GNNs have been proposed that substantially increase the expressive power, albeit at a large computational cost. Motivated by this gap, we explore alternative strategies and lower bounds. In particular, we analyze a new recursive pooling technique of local neighborhoods that allows different tradeoffs of computational cost and expressive power. First, we prove that this model can count subgraphs of size $k$, and thereby overcomes a known limitation of low-order GNNs. Second, we show how recursive pooling can exploit sparsity to reduce the computational complexity compared to the existing higher-order GNNs. More generally, we provide a (near) matching information-theoretic lower bound for counting subgraphs with graph representations that pool over representations of derived (sub-)graphs. We also discuss lower bounds on time complexity.
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