# Random graphs with given vertex degrees and switchings

## TL;DR

This paper presents a method to generate nearly uniform simple random graphs with a given degree sequence by applying switchings to the configuration model, enabling transfer of distributional results.

## Contribution

It introduces a switching-based construction that produces almost uniform simple graphs from the configuration model under bounded second moment conditions.

## Key findings

- Construction yields almost uniform distribution with $o(1)$ total variation distance.
- Method allows transfer of distributional convergence results.
- Provides applications to asymptotic normality and contiguity under degree constraints.

## Abstract

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with an almost uniform distribution, in the sense that the total variation distance is $o(1)$. This construction can be used to transfer results on distributional convergence from the configuration model multigraph to the uniform random simple graph with the given vertex degrees. As examples, we give a few applications to asymptotic normality. We show also a weaker result yielding contiguity when the maximum degree is too large for the main theorem to hold.