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

2026-01-26

AI Summary

This study investigates the limiting behavior of graph Laplacians as the internal connectivity within subgraphs tends to infinity, and establishes their effective representation on a coarse-grained graph. By employing resolvent convergence analysis in conjunction with spectral graph theory and cluster aggregation techniques, the authors prove that, for both undirected and directed graphs, the original Laplacian converges to an effective Laplacian defined on a reduced graph when intra-cluster edge weights diverge to infinity. Notably, in the directed case, the work reveals how the left and right nullspace structures associated with highly connected clusters critically shape the topology of the limiting graph. These findings extend spectral theory for directed graphs and provide a rigorous mathematical foundation for coarse-grained modeling of dynamical processes on networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of 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.

Problem

• graph Laplacian
• tightly connected clusters
• resolvent convergence
• directed graphs
• coarse-grained dynamics

Innovation

• graph Laplacian
• resolvent convergence
• coarse-graining
• directed graphs
• spectral graph theory