The Annals of Applied Probability

The asymptotic variance of the giant component of configuration model random graphs

Frank Ball and Peter Neal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a supercritical configuration model random graph, it is well known that, subject to mild conditions, there exists a unique giant component, whose size $R_{n}$ is $O(n)$, where $n$ is the total number of vertices in the random graph. Moreover, there exists $0<\rho \leq 1$ such that $R_{n}/n\stackrel{p}{\longrightarrow}\rho$ as $n\rightarrow \infty$. We show that for a sequence of well behaved configuration model random graphs with a deterministic degree sequence satisfying $0<\rho <1$; there exists $\sigma^{2}>0$, such that $\operatorname{var}(\sqrt{n}(R_{n}/n-\rho))\rightarrow \sigma^{2}$ as $n\rightarrow \infty$. Moreover, an explicit, easy to compute, formula is given for $\sigma^{2}$. This provides a key stepping stone for computing the asymptotic variance of the size of the giant component for more general random graphs.

Article information

Ann. Appl. Probab., Volume 27, Number 2 (2017), 1057-1092.

Received: March 2015
Revised: March 2016
First available in Project Euclid: 26 May 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]

Random graphs configuration model branching processes variance


Ball, Frank; Neal, Peter. The asymptotic variance of the giant component of configuration model random graphs. Ann. Appl. Probab. 27 (2017), no. 2, 1057--1092. doi:10.1214/16-AAP1225.

Export citation


  • Ball, F. and Donnelly, P. (1995). Strong approximations for epidemic models. Stochastic Process. Appl. 55 1–21.
  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • Barbour, A. D. and Mollison, D. (1990). Epidemics and random graphs. In Stochastic Processes in Epidemic Theory. Lecture Notes in Biomathematics 86 86–89. Springer, Berlin.
  • Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
  • Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124 1377–1397.
  • Britton, T., Janson, S. and Martin-Löf, A. (2007). Graphs with specified degree distributions, simple epidemics, and local vaccination strategies. Adv. in Appl. Probab. 39 922–948.
  • Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 125–145.
  • Durrett, R. (2007). Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge.
  • Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
  • Janson, S. (2009a). On percolation in random graphs with given vertex degrees. Electron. J. Probab. 14 87–118.
  • Janson, S. (2009b). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • Lefèvre, C. and Utev, S. (1999). Branching approximation for the collective epidemic model. Methodol. Comput. Appl. Probab. 1 211–228.
  • Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–179.
  • Molloy, M. and Reed, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295–305.
  • Neal, P. (2007). Coupling of two SIR epidemic models with variable susceptibilities and infectivities. J. Appl. Probab. 44 41–57.
  • Newman, M., Strogatz, S. and Watts, D. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64 026118.
  • Riordan, O. (2012). The phase transition in the configuration model. Combin. Probab. Comput. 21 265–299.
  • Waugh, W. A. O’N. (1958). Conditioned Markov processes. Biometrika 45 241–249.
  • Whittle, P. (1955). The outcome of a stochastic epidemic—a note on Bailey’s paper. Biometrika 42 116–122.