Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures is an important challenge. In this work, we generalize existing spectral clustering algorithms from static to dynamic graphs using canonical correlation analysis (CCA) to capture the temporal evolution of clusters. Based on this extended canonical correlation framework, we define the spatio-temporal graph Laplacian and investigate its spectral properties. We connect these concepts to dynamical systems theory via transfer operators, and illustrate the advantages of our method on benchmark graphs by comparison with existing methods. We show that the spatio-temporal graph Laplacian allows for a clear interpretation of cluster structure evolution over time for directed and undirected graphs.

翻译：时变图在建模复杂动态系统（如社交网络、交通流和生物过程）时频繁出现。开发技术以识别和分析这些时变图结构中的社区是一个重要挑战。在本工作中，我们利用典型相关分析（CCA）将现有的谱聚类算法从静态图推广到动态图，以捕捉聚类的时间演化。基于这一扩展的典型相关框架，我们定义了时空图拉普拉斯算子并研究了其谱特性。我们通过转移算子将这些概念与动力系统理论联系起来，并通过与现有方法的比较，在基准图上展示了我们方法的优势。我们证明，时空图拉普拉斯算子能够清晰解释有向图和无向图的聚类结构随时间演化。