Local-Global Convergence, an analytic and structural approach

TL;DR

This paper presents a unified analytic and structural framework for local-global convergence in graphs, extending the theory to unbounded degree graphs and establishing new results on clustering and quasi-limits.

Contribution

It introduces a general approach to local-global convergence, applicable to graphs with unbounded degrees, and proves the existence of modeling quasi-limits for certain graph sequences.

Findings

Extended local-global convergence to unbounded degree graphs

Proved existence of modeling quasi-limits for nowhere dense graph sequences

Enhanced understanding of clustering in local convergent sequences

Abstract

Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global convergence to graphs with unbounded degrees. As an application, we extend previous results on continuous clustering of local convergent sequences and prove the existence of modeling quasi-limits for local-global convergent sequences of nowhere dense graphs.