  A network creation game simulates a decentralized and non-cooperative
building of a communication network. Informally, there are $n$ players sitting
on the network nodes, which attempt to establish a reciprocal communication by
activating, incurring a certain cost, any of their incident links. The goal of
each player is to have all the other nodes as close as possible in the
resulting network, while buying as few links as possible. According to this
intuition, any model of the game must then appropriately address a balance
between these two conflicting objectives. Motivated by the fact that a player
might have a strong requirement about its centrality in the network, in this
paper we introduce a new setting in which if a player maintains its (either
maximum or average) distance to the other nodes within a given associated
bound, then its cost is simply equal to the number of activated edges,
otherwise its cost is unbounded. We study the problem of understanding the
structure of associated pure Nash equilibria of the resulting games, that we
call MaxBD and SumBD, respectively. For both games, we show that computing the
best response of a player is an NP-hard problem. Next, we show that when
distance bounds associated with players are non-uniform, then equilibria can be
arbitrarily bad. On the other hand, for MaxBD, we show that when nodes have a
uniform bound $R$ on the maximum distance, then the Price of Anarchy (PoA) is
lower and upper bounded by 2 and $O(n^{\frac{1}{\lfloor\log_3 R\rfloor+1}})$
for $R \geq 3$, while for the interesting case R=2, we are able to prove that
the PoA is $\Omega(\sqrt{n})$ and $O(\sqrt{n \log n})$. For the uniform SumBD
we obtain similar (asymptotically) results, and moreover we show that the PoA
becomes constant as soon as the bound on the average distance is
$n^{\omega(\frac{1}{\sqrt{\log n}})}$.
