  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..
