In this work we investigate 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 of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when $f$ is a regularly varying function at 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 $1/gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved.
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