Di-Graphs with tightly connected Clusters: Effective Graph Laplacians and Resolvent Convergence

Abstract: 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.