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It is well known that many random graphs with infinite variance degrees areultrasmall. More precisely, for configuration models and preferentialattachment models where the proportion of vertices of degree at least $k$ isapproximately $k^{-(\tau-1)}$ with $\tau\in(2,3)$, typical distances betweenpairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\logn}{|\log(\tau-2)|}$ and $\frac{4\log\log n}{|\log(\tau-2)|}$, respectively. Inthis paper, we investigate the behavior of the diameter in such models. We showthat the diameter is of order $\log\log n$ precisely when the minimal forwarddegree $d$ of vertices is at least $2$. We identify the exact constant, whichequals that of the typical distances plus $2/\log d$. Interestingly, the prooffor both models follows identical steps, even though the models are quitedifferent in nature.

Diameter in ultra-small scale-free random graphs: Extended version