How likely is an I.I.D. Degree sequence to be graphical?

Abstract

Given i.i.d. positive integer valued random variables D 1,..., D n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D 1,..., D n. We give sufficient conditions on the distribution of D i for the probability that this be the case to be asymptotically 0, 1/2 or strictly between 0 and 1/2. These conditions roughly correspond to whether the limit of nP(D i ≥ n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs. © Institute of Mathematical Statistics, 2005.