#### Bounded-Distance Network Creation Games

##### Davide Bilò, Luciano Gualà, Guido Proietti

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 $n^{\omega(\frac{1}{\sqrt{\log n}})}$.

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