# Recovering the Structure of Random Linear Graphs

## TL;DR

This paper presents a spectral method for reconstructing the linear order of vertices in random linear graphs, providing tight bounds on reconstruction accuracy and vertex placement errors.

## Contribution

It introduces a novel spectral approach leveraging random matrix theory to recover vertex order in random linear graphs, with proven bounds on accuracy.

## Key findings

- Successfully recovers vertex order in simulated graphs
- Provides tight bounds on misplaced vertices
- Quantifies drift from natural positions

## Abstract

In a random linear graph, vertices are points on a line, and pairs of vertices are connected, independently, with a link probability that decreases with distance. We study the problem of reconstructing the linear embedding from the graph, by recovering the natural order in which the vertices are placed. We propose an approach based on the spectrum of the graph, using recent results on random matrices. We demonstrate our method on a particular type of random linear graph. We recover the order and give tight bounds on the number of misplaced vertices, and on the amount of drift from their natural positions.