Diameter of P.A. random graphs with edge-step functions

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function $f:\mathbb{N}\to [0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when $f$ is a $\mathit{\text{regularly}}\mathit{\text{varying}}\mathit{\text{function}}\mathit{\text{at}}\mathit{\text{infinity}}$ with strictly negative index of regular variation~$-\gamma$. For this particular class, we prove a characterization for the diameter that depends only on~$-\gamma$. More specifically, we prove that the diameter of such graphs is of order $1\mathrm{/}\gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of \textit{slowly varying functions} are also obtained.