# Asymptotic normality in random graphs with given vertex degrees

## TL;DR

This paper proves that the number of specific tree components in random graphs with a fixed degree sequence follows a normal distribution asymptotically, using the method of moments and cumulant estimates.

## Contribution

It establishes asymptotic normality for component counts in both multigraphs and simple graphs with given degrees, extending previous results and providing new proofs.

## Key findings

- Asymptotic normality of component counts proven for both multigraphs and simple graphs.
- New cumulant estimates for weakly dependent variables.
- Extended results on the size of the giant component in random graphs.

## Abstract

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and R\"ollin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.