Distances in random graphs with finite mean and infinite variance degrees

In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent $τ\in (2,3)$. The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. When $τ\in (2,3)$, this graph distance grows like $2\frac{\log\log N}{|\log(τ-2)|}$. In different papers, the cases $τ>3$ and $τ\in (1,2)$ have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.

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