Distances in Random Graphs with Finite Mean and Infinite Variance Degrees

In this paper we study typical distances in random graphs with i.i.d. degrees of which the tail of the common distribution function is regularly varying with exponent $1-\tau$. Depending on the value of the parameter $\tau$ we can distinct three cases: (i) $\tau>3$, where the degrees have finite variance, (ii) $\tau\in (2,3)$, where the degrees have infinite variance, but finite mean, and (iii) $\tau\in (1,2)$, where the degrees have infinite mean. The distances between two randomly chosen nodes belonging to the same connected component, for $\tau>3$ and $\tau \in (1,2),$ have been studied in previous publications, and we survey these results here. When $\tau\in (2,3)$, the graph distance centers around $2\log\log{N}/|\log(\tau-2)|$. We present a full proof of this result, and study the fluctuations around this asymptotic means, by describing the asymptotic distribution. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only. The random graphs studied here can serve as models for complex networks where degree power laws are observed; this is illustrated by comparing the typical distance in this model to Internet data, where a degree power law with exponent $\tau\approx 2.2$ is observed for the so-called Autonomous Systems (AS) graph