Distances in random graphs with infinite mean degrees

Abstract

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function $F$ is regularly varying with exponent $τ\in (1,2)$. Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. The paper is part of a sequel of three papers. The other two papers study the case where $τ\in (2,3)$, and $τ\in (3,\infty),$ respectively. The main result of this paper is that the graph distance converges for $τ\in (1,2)$ to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most $N^α$ for some $α>0.$ For $τ^{-1}<α<(τ-1)^{-1}$, the hopcount converges to 3 in probability, while for $α>(τ-1)^{-1}$, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.