We propose a random graph model with preferential attachment rule and emph{edge-step functions} that govern the growth rate of the vertex set. We study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function $f$ is a emph{monotone regularly varying function} at infinity, the sequence of graphs associated to it obeys a power-law degree distribution whose exponent is related to the index of regular variation of $f$ at infinity whenever said index is greater than $-1$. When the regularly variation index is less than or equal to $-1$, we show that the proportion of vertices with degree smaller than any given constant goes to $0$ a. s..
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