  We study Nash equilibria and the price of anarchy in the classic model of
Network Creation Games introduced by Fabrikant et al. In this model every agent
(node) buys links at a prefixed price $\alpha > 0$ in order to get connected to
the network formed by all the $n$ agents. In this setting, the reformulated
tree conjecture states that for $\alpha > n$, every Nash equilibrium network is
a tree. Moreover, Demaine et al. conjectured that the price of anarchy for this
model is constant. Since it was shown that the price of anarchy for trees is
constant, if the tree conjecture were true, then the price of anarchy would be
constant for $\alpha > n$.
  Up to now it has been proved that the \PoA is constant $(i)$ in the
\emph{lower range}, for $\alpha = O(n^{1-\delta})$ with $\delta \geq
\frac{1}{\log n}$ and $(ii)$ in the \emph{upper range}, for $\alpha > 4n-13$.
In contrast, the best upper bound known for the price of anarchy for the
remaining range is $2^{O(\sqrt{\log n})}$.
  In this paper we give new insights into the structure of the Nash equilibria
for $\alpha > n$ and we enlarge the range of the parameter $\alpha$ for which
the price of anarchy is constant. Specifically, we prove that the price of
anarchy is constant for $\alpha > n(1+\epsilon)$ by showing that every
equilibrium of diameter greater than some prefixed constant is a tree.
