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。受现实应用中大规模数据集普遍存在的驱动，池化层成为GNNs的关键组成部分。通过缩减输入图的规模，池化能够加速训练过程并可能提升泛化能力。然而，现有池化操作常以牺牲基础图结构为代价来优化学习任务，从而降低了模型的可解释性。这导致其在不同数据集类型、下游任务及池化比率下的性能表现不稳定。针对这些问题，我们提出通过边折叠实现结构感知池化的新型图池化层。我们的方法利用扩散几何原理，在保持图度量结构及结构多样性的同时迭代缩减图的规模。我们采用等距不变多样性度量指标——量级（magnitude）来指导池化过程，从而实现对池化过程保真度的控制。此外，我们引入度量空间的展度（spread）作为更快速、更稳定的替代方案，以确保计算效率。实验结果表明，我们的方法（i）在多种图分类任务中相较其他池化层达到最优性能，（ii）保留了输入图的关键谱特性，且（iii）在不同池化比率下均保持较高的准确率。