  We introduce a novel preferential attachment model using the draw variables
of a modified P\'olya urn with an expanding number of colors, notably capable
of modeling influential opinions (in terms of vertices of high degree) as the
graph evolves. Similar to the Barab\'asi-Albert model, the generated graph
grows in size by one vertex at each time instance; in contrast however, each
vertex of the graph is uniquely characterized by a color, which is represented
by a ball color in the P\'olya urn. More specifically at each time step, we
draw a ball from the urn and return it to the urn along with a number
(potentially time-varying and non-integer) of reinforcing balls of the same
color; we also add another ball of a new color to the urn. We then construct an
edge between the new vertex (corresponding to the new color) and the existing
vertex whose color ball is drawn. Using color-coded vertices in conjunction
with the time-varying reinforcing parameter allows for vertices added (born)
later in the process to potentially attain a high degree in a way that is not
captured in the Barab\'asi-Albert model. We study the degree count of the
vertices by analyzing the draw vectors of the underlying stochastic process. In
particular, we establish the probability distribution of the random variable
counting the number of draws of a given color which determines the degree of
the vertex corresponding to that color in the graph. We further provide
simulation results presenting a comparison between our model and the
Barab\'asi-Albert network.
