Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-world applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation. However, existing pooling operations often optimise for the learning task at the expense of discarding fundamental graph structures, thus reducing interpretability. This leads to unreliable performance across dataset types, downstream tasks and pooling ratios. Addressing these concerns, we propose novel graph pooling layers for structure-aware pooling via edge collapses. Our methods leverage diffusion geometry and iteratively reduce a graph's size while preserving both its metric structure and its structural diversity. We guide pooling using magnitude, an isometry-invariant diversity measure, which permits us to control the fidelity of the pooling process. Further, we use the spread of a metric space as a faster and more stable alternative ensuring computational efficiency. Empirical results demonstrate that our methods (i) achieve top performance compared to alternative pooling layers across a range of diverse graph classification tasks, (ii) preserve key spectral properties of the input graphs, and (iii) retain high accuracy across varying pooling ratios.

翻译：图神经网络（GNNs）在图相关任务中已展现出显著成效。鉴于现实应用中海量数据集的普遍存在，池化层成为GNNs的关键组件。通过缩减输入图的规模，池化操作能加速训练过程并可能提升泛化能力。然而，现有池化方法常为优化学习任务而牺牲基础图结构，导致可解释性降低，进而在不同数据集类型、下游任务及池化比率下产生不可靠的性能表现。针对这些问题，我们提出基于边坍缩的结构感知图池化新方法。该技术利用扩散几何原理，在迭代缩减图规模的同时保持其度量结构与结构多样性。我们采用具有等距不变性的多样性度量指标——幅度（magnitude）来指导池化过程，从而控制池化操作的保真度。此外，我们引入度量空间的展宽（spread）作为更快速稳定的替代方案以确保计算效率。实验结果表明，我们的方法具有以下优势：（i）在多种图分类任务中相较其他池化层达到最优性能；（ii）保持输入图的关键谱特性；（iii）在不同池化比率下均保持高精度。