  This paper presents a tighter bound on the degree distribution of arbitrary
P\'{o}lya urn graph processes, proving that the proportion of vertices with
degree $d$ obeys a power-law distribution $P(d) \propto d^{-\gamma}$ for $d
\leq n^{\frac{1}{6}-\epsilon}$ for any $\epsilon > 0$, where $n$ represents the
number of vertices in the network. Previous work by Bollob\'{a}s et al.
formalized the well-known preferential attachment model of Barab\'{a}si and
Albert, and showed that the power-law distribution held for $d \leq
n^{\frac{1}{15}}$ with $\gamma = 3$. Our revised bound represents a significant
improvement over existing models of degree distribution in scale-free networks,
where its tightness is restricted by the Azuma-Hoeffding concentration
inequality for martingales. We achieve this tighter bound through a careful
analysis of the first set of vertices in the network generation process, and
show that the newly acquired is at the edge of exhausting Bollob\'as model in
the sense that the degree expectation breaks down for other powers.
