|Résumé||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.