Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying node embeddings, whose usefulness depends on well defined stability properties across nodes and across time. We introduce Unfolded Laplacian Spectral Embedding (ULSE), a principled extension of unfolded adjacency spectral embedding to normalized Laplacian operators, a setting where stability guarantees have remained out of reach. We prove that ULSE satisfies both cross-sectional and longitudinal stability under a dynamic stochastic block model. Moreover, the Laplacian formulation yields a dynamic Cheeger-type inequality linking the spectrum of the unfolded normalized Laplacian to worst case conductance over time, providing structural insight into the embeddings. Empirical results on synthetic and real world dynamic networks validate the theory.

翻译：动态关系数据广泛存在于机器学习应用中，但其时变结构对学习具有时间一致性与可解释性的表示提出了挑战。常见方法是学习时变节点嵌入，其有效性取决于节点间及时序上明确定义的稳定性性质。本文提出展开拉普拉斯谱嵌入（ULSE），该方法将展开邻接谱嵌入原则性扩展至归一化拉普拉斯算子——这一设定下的稳定性保证此前尚未实现。我们证明在动态随机块模型下，ULSE同时满足截面稳定性与纵向稳定性。此外，拉普拉斯公式导出了一个动态Cheeger型不等式，将展开归一化拉普拉斯的谱与时间维度上的最差传导率相关联，从而为嵌入提供了结构层面的理论解释。在合成与真实动态网络上的实证结果验证了该理论。