In this note, we study Laplacians on graphs for which connectivity within certain sub-graphs 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 agregating 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：本文研究一类特殊图上的拉普拉斯算子，其特定子图内部的连接性趋于无穷。我们主要关注共享公共节点集的图，其中某些簇内部的边权重趋于无穷。随着簇内连接性的增强，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，其中每个高度连接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接性增强的具体方式，其对应效应由限制在高连接性簇上的拉普拉斯算子的左核与右核结构所中介。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。