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Scaling limits of random graphs

Louigi Addario-Berry, Christina Goldschmidt

Abstract

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric structure of these random objects in the limit as their size goes to infinity, with distances appropriately rescaled. We start with the simplest setting of random trees, before turning to various examples of random graphs, including the critical Erdős--Rényi random graph.

Scaling limits of random graphs

Abstract

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric structure of these random objects in the limit as their size goes to infinity, with distances appropriately rescaled. We start with the simplest setting of random trees, before turning to various examples of random graphs, including the critical Erdős--Rényi random graph.

Paper Structure

This paper contains 10 sections, 42 equations.

Table of Contents

  1. Introduction
  2. Models of random trees and graphs
  3. Connected graphs as metric spaces, and convergence
  4. The contour process of a tree
  5. Line-breaking constructions of trees
  6. The depth-first queue process of a tree
  7. Graphs: cores, kernels and line-breaking constructions
  8. Constructing a graph from the depth-first queue process
  9. The critical Erdős--Rényi random graph
  10. Uniform random graphs with i.i.d. degrees