  Selfish Network Creation focuses on modeling real world networks from a
game-theoretic point of view. One of the classic models by Fabrikant et al.
[PODC'03] is the network creation game, where agents correspond to nodes in a
network which buy incident edges for the price of $\alpha$ per edge to minimize
their total distance to all other nodes. The model is well-studied but still
has intriguing open problems. The most famous conjectures state that the price
of anarchy is constant for all $\alpha$ and that for $\alpha \geq n$ all
equilibrium networks are trees.
  We introduce a novel technique for analyzing stable networks for high
edge-price $\alpha$ and employ it to improve on the best known bounds for both
conjectures. In particular we show that for $\alpha > 4n-13$ all equilibrium
networks must be trees, which implies a constant price of anarchy for this
range of $\alpha$. Moreover, we also improve the constant upper bound on the
price of anarchy for equilibrium trees.
