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Critical Window for Connectivity in the Configuration Model

Part of: Graph theory Combinatorial probability 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: l.federico@tue.nl, r.w.v.d.hofstad@tue.nl)
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: l.federico@tue.nl, r.w.v.d.hofstad@tue.nl)
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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 

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