Network Creation Games: Structure vs Anarchy.

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 $\alpha 4$, induces an $\epsilon-$distance-almost-uniform graph.