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

TL;DR

This paper investigates how graph Laplacians behave when intra-cluster connectivity becomes very high, showing they converge to a simplified Laplacian on a coarsened graph, which aids understanding of large-scale network dynamics.

Contribution

It introduces a rigorous analysis of resolvent convergence of graph Laplacians with increasing intra-cluster weights, extending spectral graph theory to directed graphs and cluster aggregation.

Findings

Laplacians converge to an effective Laplacian on a coarsened graph as intra-cluster weights grow.

Effective Laplacian captures the limiting behavior of highly connected clusters.

Results illuminate coarse-grained dynamics in large-scale directed networks.

Abstract

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.