Random graphs with arbitrary i.i.d. degrees

Abstract

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree distribution is regularly varying with exponent $1-τ$. There are three distinct cases: (i) $τ>3$, where the degrees have finite variance, (ii) $τ\in (2,3)$, where the degrees have infinite variance, but finite mean, and (iii) $τ\in (1,2)$, where the degrees have infinite mean. These random graphs can serve as models for complex networks where degree power laws are observed. Our results are twofold. First, we give a criterion when there exists a unique largest connected component of size proportional to the size of the graph, and study sizes of the other connected components. Secondly, we establish a phase transition for the diameter when $τ\in (2,3)$. Indeed, we show that for $τ>2$ and when nodes with degree 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are at least 3 or more, we show that, for $τ\in (2,3)$, the diameter of the graph is with high probability bounded from above by a constant times the $\log \log$ of the size of the graph.