Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit laws for self-loops and multiple edges in the configuration model

Omer Angel, Remco van der Hofstad, and Cecilia Holmgren

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Abstract

We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges.

We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide estimations on the total variation distance between the numbers of self-loops and multiple edges and their limits, as well as between the sum of these values and the Poisson random variable to which this sum converges to. This revisits previous works of Bollobás, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees.

The error estimates follow from an application of the Stein–Chen method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramér–Wold device using thinning, which is of independent interest.

When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph. Our results and proofs easily extend to directed and bipartite configuration models.

Résumé

Nous considérons les boucles et les arêtes multiples dans le modèle de configuration lorsque la taille du graphe tend vers l’infini. L’intérêt de ces variables aléatoires est dû au fait que le modèle de configuration, conditionné à la simplicité, est un graphe aléatoire uniforme avec des degrés prescrits. La simplicité correspond à l’absence des boucles et des arêtes multiples.

Nous montrons que le nombre des boucles et des arêtes multiples converge en loi vers deux variables aléatoires indépendantes qui suivent des lois de Poisson lorsque le moment d’ordre 2 de la loi empirique des degrés converge. Nous fournissons aussi des estimations des distances de variation totale entre les nombres des boucles et des arêtes multiples et leurs limites, ainsi qu’entre la somme de ces nombres et la variable aléatoire, qui suit une loi de Poisson, vers laquelle converge cette somme. Cela revisite les œuvres précédentes de Bollobás comme de Janson, de Wormald, et d’autres. Les estimations d’erreur impliquent également une asymptotique précise pour le nombre de graphes simples avec des degrés prescrits.

Les estimations d’erreur découlent d’une application de la méthode de Stein–Chen pour la convergence vers une loi de Poisson, qui est une nouvelle méthode pour ce problème. L’indépendance asymptotique des boucles et des arêtes multiples suit à partir d’une version Poisson du dispositif Cramér–Wold utilisant l’amincissement, qui est intéressant en lui-même.

Lorsque la loi des degrés a un moment d’ordre 2 infini, nos résultats généraux échouent. Nous pouvons, cependant, prouver un théorème de la limite centrale pour le nombre des boucles, et pour les arêtes multiples entre sommets avec degrés beaucoup plus petits que la racine carrée de la taille du graphe. Nos résultats et preuves peuvent facilement s’étendre aux modèles de configuration orientés et bipartis.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1509-1530.

Dates
Received: 2 February 2017
Revised: 11 June 2018
Accepted: 21 August 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398877

Digital Object Identifier
doi:10.1214/18-AIHP926

Mathematical Reviews number (MathSciNet)
MR4010943

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82B43: Percolation [See also 60K35]

Keywords
Configuration model Self-loops Multiple edges Chen–Stein Poisson approximation

Citation

Angel, Omer; van der Hofstad, Remco; Holmgren, Cecilia. Limit laws for self-loops and multiple edges in the configuration model. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1509--1530. doi:10.1214/18-AIHP926. https://projecteuclid.org/euclid.aihp/1569398877


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