# Large, lengthy graphs look locally like lines

## TL;DR

This paper uses unimodular random rooted graph theory to show that large bounded degree graphs with proportional diameter and volume have many vertices where the local structure resembles a line segment at a mesoscopic scale.

## Contribution

It introduces a novel application of unimodular graph theory to analyze the local geometric structure of large graphs, revealing line-like behavior at certain scales.

## Key findings

- Many vertices exhibit local geometry close to a line segment
- Large graphs have mesoscopic scales where they resemble $
eal$
- The graph's metric structure is characterized in terms of Gromov-Hausdorff convergence

## Abstract

We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like $\mathbb{R}$ in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric.