New Insights into the Structure of Equilibria for the Network Creation Game

We study the sum classic network creation game introduced by Fabrikant et al. in which $n$ players conform a network buying links at individual price $\alpha$. When studying this model we are mostly interested in \emph{Nash equilibria} (\NE) and the \emph{Price of Anarchy} (\PoA). It is conjectured that the \PoA is constant for any $\alpha$. Up until now, it has been proved constant \PoA for the range $\alpha = O(n^{1-\delta_1})$ with $\delta_1>0$ a positive constant and it has been proved constant \PoA for the range $\alpha >n(1+\delta_2)$ with $\delta_2>0$ a positive constant. Our contribution consists in proving that \NE graphs satisfy very restrictive topological properties generalising some properties proved in the literature and providing new insights that might help settling the conjecture that the \PoA is constant for the remaining range of $\alpha$: (i) We show that \emph{every} node has the majority of the other nodes of the \NE graph at the \emph{same} narrow range of distances. (ii) If instead of considering the average degree, we focus on the number of non-bridge links bought for any player, we can prove that there exists a constant $D$ such that the average degree is upper bounded by $\max(R, \frac{2n}{\alpha}+6)$ for any \NE graph of diameter larger than $D$, where $R$ is a positive constant.