Skip to main content Accessibility help
×
Home
Hostname: page-component-5bf98f6d76-gckwl Total loading time: 12.8 Render date: 2021-04-21T08:10:37.998Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Critical Window for Connectivity in the Configuration Model

Published online by Cambridge University Press:  29 May 2017

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)
Corresponding

Abstract

We identify the asymptotic probability of a configuration model CM n (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.

Type
Paper
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.

References

[1] Arratia, R. and Gordon, L. (1989) Tutorial on large deviations for the binomial distribution. Bull. Math. Biol. 51 125131.CrossRefGoogle ScholarPubMed
[2] Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar
[3] van der Hofstad, R. (2017) Random Graphs and Complex Networks, Vol. 1, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.CrossRefGoogle Scholar
[4] Janson, S. (2008) The largest component in a subcritical random graph with a power law degree distribution. Ann. Appl. Probab. 18 16511668.CrossRefGoogle Scholar
[5] Janson, S. (2009) The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205225.CrossRefGoogle Scholar
[6] Janson, S. (2014) The probability that a random multigraph is simple, II. J. Appl. Probab. 51A 123137.CrossRefGoogle Scholar
[7] Janson, S. and Luczak, M. J. (2009) A new approach to the giant component problem. Random Struct. Alg. 34 197216.CrossRefGoogle 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. 165182.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 161179.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 295305.CrossRefGoogle Scholar
[12] Wormald, N. C. (1981) The asymptotic connectivity of labelled regular graphs. J. Combin. Theory Ser. B 31 156167.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 83 *
View data table for this chart

* Views captured on Cambridge Core between 29th May 2017 - 21st April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Critical Window for Connectivity in the Configuration Model
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Critical Window for Connectivity in the Configuration Model
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Critical Window for Connectivity in the Configuration Model
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *