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

TL;DR

This work develops a resolvent-based coarse-graining framework for graph Laplacians with strongly connected clusters. By analyzing both undirected and directed graphs, it shows that as intra-cluster connectivity diverges, the original Laplacians converge to effective Laplacians on a reduced graph: in the undirected case via strong/norm resolvent convergence to $\underline{L}$ with $L_\beta \to J^\uparrow \underline{L} J^\downarrow$, and in the directed case via non-selfadjoint analysis using the Riesz projector to obtain reduced operators $\underline{L}^-$ and $\underline{L}^+$ on coarse graphs $\underline{G}^-$ and $\underline{G}^+$. The results rely on precise kernel structures of $L^{-}$ and $L^{+}$, define projection- and interpolation-operators $J_{\downarrow}, J_{\uparrow}$ (and their directed analogues), and provide explicit mass/edge-weight aggregation formulas for the coarse graphs. They yield norm-resolvent convergence with rates $\mathcal{O}(1/\beta)$ in finite-cluster scenarios and imply convergence of heat kernels via resolvent convergence, thereby giving a rigorous basis for coarse-grained dynamics and spectral graph analysis in both undirected and 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.

Figures (11)

• Figure 1: Example graphs with different reaches on the same node set
• Figure 2: Example graphs of Fig \ref{['weight_vector_example']}. with reversed edge directions
• Figure 3: Original graph $G$ and reduced graph $\underline{G}$.
• Figure 4: Path graph $G$
• Figure 5: Reduced path graph $\underline{G}$

Theorems & Definitions (75)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.6
• proof
• Definition 2.7
• Proposition 2.1
• proof
• Proposition 2.8