In this note, we study Laplacians on graphs for which connectivity within certain subgraphs tends to infinity. Our main focus are graphs sharing a common node set on which edge weights within certain clusters grow to infinity. As intra-cluster connectivity increases, we show that the corresponding graph Laplacians converge -- in the resolvent sense -- to an effective graph Laplacian. This effective limit Laplacian is defined on a coarsened graph, where each highly connected cluster is collapsed into a single node. In the undirected setting, the effective Laplacian arises naturally from aggregating over tightly connected clusters. In the directed case, the limiting graph structure depends on the precise manner in which connectivity increases; with the corresponding effects mediated by the left and right kernel structure of the Laplacian restricted to high-connectivity clusters. Our results shed light on the emergence of coarse-grained dynamics in large-scale networks and contribute to spectral graph theory of directed graphs.

翻译：本文研究特定子图内部连通性趋于无穷大时图上的拉普拉斯算子。我们主要关注共享公共节点集的图，其中特定簇内的边权重趋于无穷大。随着簇内连通性的增强，我们证明相应的图拉普拉斯算子在预解意义下收敛于一个有效图拉普拉斯算子。该有效极限拉普拉斯算子定义在粗化图上，其中每个高度连通的簇被坍缩为单一节点。在无向图情形中，有效拉普拉斯算子通过紧密连接簇的聚合自然产生。在有向图情形中，极限图结构取决于连通性增长的具体方式，其相应效应由限制在高连通性簇上的拉普拉斯算子的左、右核结构所中介。我们的研究结果揭示了大规模网络中粗粒度动力学的涌现机制，并为有向图的谱图理论作出了贡献。