Limits of local-global convergent graph sequences

TL;DR

This paper investigates the limits of sparse graph sequences under local-global convergence, showing they can be represented by graphings and exploring related concepts like Bernoulli graphings and hyperfiniteness.

Contribution

It proves that limits of bounded degree graph sequences under local-global convergence can be represented by graphings, extending previous notions and analyzing related structures.

Findings

Limit objects are representable by graphings

Analysis of Bernoulli graphings and factor of i.i.d. processes

Insights into hyperfiniteness in sparse graph limits

Abstract

The colored neighborhood metric for sparse graphs was introduced by Bollob\'as and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.