The Annals of Applied Probability

How likely is an i.i.d. degree sequence to be graphical?

Richard Arratia and Thomas M. Liggett

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Given i.i.d. positive integer valued random variables D1,…,Dn, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D1,…,Dn. We give sufficient conditions on the distribution of Di for the probability that this be the case to be asymptotically 0, ½ or strictly between 0 and ½. These conditions roughly correspond to whether the limit of nP(Din) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 652-670.

First available in Project Euclid: 1 February 2005

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Zentralblatt MATH identifier

Primary: 05C07: Vertex degrees [See also 05E30] 05C80: Random graphs [See also 60B20] 60G70: Extreme value theory; extremal processes

Simple graphs random graphs degree sequences extremes of i.i.d. random variables


Arratia, Richard; Liggett, Thomas M. How likely is an i.i.d. degree sequence to be graphical?. Ann. Appl. Probab. 15 (2005), no. 1B, 652--670. doi:10.1214/105051604000000693.

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