Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than PH and spectral information on graphs alone. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.

翻译：引入复杂的拓扑信息（如环结构）已被证明能增强消息传递图神经网络（GNNs）的表达能力，使其超越Weisfeiler-Leman（WL）层次。因此，持久同调（PH）方法正日益广泛地应用于图表示学习。在此背景下，近期研究提出通过顶点和边特征来增强经典PH图，以提升表达能力。然而，这些方法仍未能捕捉基本的图结构信息。本文提出SpectRe——一种新的图拓扑描述符，它将谱信息整合到PH图中。值得注意的是，SpectRe的表达能力严格超越了单独的PH与图谱信息。我们还引入了全局与局部稳定性的概念来分析现有描述符，并证明SpectRe具有局部稳定性。最后，在合成与真实数据集上的实验验证了SpectRe的有效性及其在相关学习任务中增强图模型能力的潜力。代码发布于https://github.com/Aalto-QuML/SpectRe/。