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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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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]

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


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.

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  • [1] R. Albert and A.-L. Barabási. Statistical mechanics of complex networks. Rev. Modern Phys. 74 (1) (2002) 47–97.
  • [2] N. Alon and J. Spencer. The Probabilistic Method, 2nd edition. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 2000.
  • [3] A. Barbour, L. Holst and S. Janson. Poisson Approximation. Oxford Studies in Probability. 2. The Clarendon Press Oxford University Press, New York, 1992.
  • [4] E. A. Bender and E. R. Canfield. The asymptotic number of labelled graphs with given degree sequences. J. Combin. Theory Ser. A 24 (1978) 296–307.
  • [5] J. Blanchet and A. Stauffer. Characterizing optimal sampling of binary contingency tables via the configuration model. Random Structures Algorithms 42 (2) (2013) 159–184.
  • [6] B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 (4) (1980) 311–316.
  • [7] B. Bollobás. Random Graphs, 2nd edition. Cambridge Studies in Advanced Mathematics 73. Cambridge University Press, Cambridge, 2001.
  • [8] T. Britton, M. Deijfen and A. Martin-Löf. Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124 (6) (2006) 1377–1397.
  • [9] C. Cooper and A. Frieze. The size of the largest strongly connected component of a random digraph with a given degree sequence. Combin. Probab. Comput. 13 (3) (2004) 319–337.
  • [10] P. Gao and N. Wormald. Enumeration of graphs with a heavy-tailed degree sequence. Adv. Math. 287 (2016) 412–450.
  • [11] C. Holmgren and S. Janson. Using Stein’s method to show Poisson and normal limit laws for fringe subtrees. Discrete Math. Theor. Comput. Sci. BA (2014) 169–180.
  • [12] C. Holmgren and S. Janson. Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Probab. 20 (4) (2015) 1–51.
  • [13] S. Janson. The probability that a random multigraph is simple. Combin. Probab. Comput. 18 (1–2) (2009) 205–225.
  • [14] S. Janson. The probability that a random multigraph is simple. II. J. Appl. Probab. 51A (2014) 123–137.
  • [15] S. Janson, M. Luczak and P. Windridge. Law of large numbers for the sir epidemic on a random graph with given degrees. Random Structures Algorithms 45 (4) (2014) 724–761.
  • [16] B. D. McKay and N. C. Wormald. Asymptotic enumeration by degree sequence of graphs with degrees $o(n^{1/2})$. Combinatorica 11 (4) (1991) 369–382.
  • [17] M. Molloy and B. Reed. A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 (2–3) (1995) 161–179.
  • [18] M. Molloy and B. Reed. The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 (3) (1998) 295–305.
  • [19] M. E. J. Newman. The structure and function of complex networks. SIAM Rev. 45 (2) (2003) 167–256 (electronic).
  • [20] M. E. J. Newman. Random graphs as models of networks. In Handbook of Graphs and Networks 35–68. Wiley-VCH, Weinheim, 2003.
  • [21] M. E. J. Newman, S. Strogatz and D. Watts. Random graphs with arbitrary degree distribution and their application. Phys. Rev. E 64 (2000) 026118.
  • [22] R. van der Hofstad. Random Graphs and Complex Networks. Cambridge Series in Statistical and Probabilistic Mathematics 1. Cambridge University Press, Cambridge, 2017.
  • [23] R. van der Hofstad and J. Komjáthy. When is a scale-free graph ultra-small?. Journ. Stat. Phys. 169 (2017) 223–264.
  • [24] P. van der Hoorn and N. Litvak. Upper bounds for number of removed edges in the erased configuration model. In Algorithms and Models for the Web Graph 54–65. Lecture Notes in Comput. Sci. 9479. Springer, Cham, 2015.
  • [25] N. C. Wormald. The asymptotic distribution of short cycles in random regular graphs. J. Combin. Theory Ser. B 31 (2) (1981) 168–182.