Graphs and networks play an important role in modeling and analyzing complex interconnected systems such as transportation networks, integrated circuits, power grids, citation graphs, and biological and artificial neural networks. Graph clustering algorithms can be used to detect groups of strongly connected vertices and to derive coarse-grained models. We define transfer operators such as the Koopman operator and the Perron-Frobenius operator on graphs, study their spectral properties, introduce Galerkin projections of these operators, and illustrate how reduced representations can be estimated from data. In particular, we show that spectral clustering of undirected graphs can be interpreted in terms of eigenfunctions of the Koopman operator and propose novel clustering algorithms for directed graphs based on generalized transfer operators. We demonstrate the efficacy of the resulting algorithms on several benchmark problems and provide different interpretations of clusters.

翻译：图和网络在建模与分析复杂互联系统中扮演着重要角色，例如交通网络、集成电路、电网、引文图谱以及生物与人工神经网络。图聚类算法可用于检测强连通顶点群组并推导粗粒度模型。我们在图上定义了迁移算子（如库普曼算子和佩龙-弗罗贝尼乌斯算子），研究了它们的谱性质，引入了这些算子的伽辽金投影，并阐释了如何从数据中估计降阶表示。特别地，我们证明了无向图的谱聚类可解释为库普曼算子特征函数的体现，并基于广义迁移算子提出了针对有向图的新型聚类算法。我们在多个基准问题上验证了所提算法的有效性，并提供了对聚类的多重解释。