  We study Nash equilibria and the price of anarchy in the classic model of
Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou
and Shenker in 2003. This is a selfish network creation model where players
correspond to nodes in a network and each of them can create links to the other
$n-1$ players at a prefixed price $\alpha > 0$. The player's goal is to
minimise the sum of her cost buying edges and her cost for using the resulting
network. One of the main conjectures for this model states that the price of
anarchy, i.e. the relative cost of the lack of coordination, is constant for
all $\alpha$. This conjecture has been confirmed for $\alpha = O(n^{1-\delta})$
with $\delta \geq 1/\log n$ and for $\alpha > 4n-13$. The best known upper
bound on the price of anarchy for the remaining range is $2^{O(\sqrt{\log
n})}$.
  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 for any small $\epsilon>0$,
the price of anarchy is constant for $\alpha > n(1+\epsilon)$ by showing that
any biconnected component of any non-trivial Nash equilibrium, if it exists,
has at most a constant number of nodes.
