  We study Nash equilibria and the price of anarchy in the classical 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. 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
$\alpha >n$. Moreover, Demaine et al. conjectured that the price of anarchy for
this model is constant.
  Up to now the last conjecture has been proven in (i) the \emph{lower range},
for $\alpha = O(n^{1-\epsilon})$ with $\epsilon \geq \frac{1}{\log n}$ and (ii)
in the \emph{upper range}, for $\alpha > 65n$. 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 different ranges of $\alpha$ and we enlarge the range for which the price
of anarchy is constant. Regarding the upper range, we prove that every Nash
equilibrium is a tree for $\alpha > 17n$ and that the price of anarchy is
constant even for $\alpha > 9n$. In the lower range, we show that any Nash
equilibrium for $\alpha < n/C$ with $C > 4$, induces an
$\epsilon-$distance-almost-uniform graph.
