Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T21:41:25.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical Window for Connectivity in the Configuration Model

Part of: Combinatorial probability Graph theory Combinatorics

Published online by Cambridge University Press:  29 May 2017

LORENZO FEDERICO  and
REMCO VAN DER HOFSTAD
LORENZO FEDERICO
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected], [email protected])
REMCO VAN DER HOFSTAD
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

MSC classification

Primary: 05C80: Random graphs 05A16: Asymptotic enumeration
Secondary: 05C30: Enumeration in graph theory 05C40: Connectivity 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 660 - 680
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arratia, R. and Gordon, L. (1989) Tutorial on large deviations for the binomial distribution. Bull. Math. Biol. 51 125–131.Google Scholar
[2] Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.Google Scholar
[3] van der Hofstad, R. (2017) Random Graphs and Complex Networks, Vol. 1, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.Google Scholar
[4] Janson, S. (2008) The largest component in a subcritical random graph with a power law degree distribution. Ann. Appl. Probab. 18 1651–1668.Google Scholar
[5] Janson, S. (2009) The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.Google Scholar
[6] Janson, S. (2014) The probability that a random multigraph is simple, II. J. Appl. Probab. 51A 123–137.Google Scholar
[7] Janson, S. and Luczak, M. J. (2009) A new approach to the giant component problem. Random Struct. Alg. 34 197–216.Google Scholar
[8] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience.CrossRefGoogle Scholar
[9] Łuczak, T. (1992) Sparse random graphs with a given degree sequence. In Random Graphs (Poznań 1989), Vol. 2, Wiley-Interscience, pp. 165–182.Google Scholar
[10] Molloy, M. and Reed, B. (1995) A critical point for random graphs with a given degree sequence. In Proc. Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science: Random Graphs '93 (Poznań 1993), Random Struct. Alg. 6 161–179.CrossRefGoogle Scholar
[11] 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.CrossRefGoogle Scholar
[12] Wormald, N. C. (1981) The asymptotic connectivity of labelled regular graphs. J. Combin. Theory Ser. B 31 156–167.CrossRefGoogle Scholar