Basic Network Creation Games

We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes, by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they globally minimize diameter. For the local-average-distance version, we prove an upper bound of 2O(√ lg n), a lower bound of 3, a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the local-diameter version, we prove a lower bound of Ω(√ n), and a tight upper bound of 3 for trees. All of our upper bounds apply equally well to previously extensively studied network creation games, both in terms of the diameter metric described above and the previously studied price of anarchy (which are related by constant factors). In surprising contrast, our model has no parameter α for the link creation cost, so our results automatically apply for all values of alpha without additional effort; furthermore, equilibrium can be checked in polynomial time in our model, unlike previous models. Our perspective enables simpler and more general proofs that get at the heart of network creation games

On the Structure of Equilibria in Basic Network Formation

We study network connection games where the nodes of a network perform edge swaps in order to improve their communication costs. For the model proposed by Alon et al. (2010), in which the selfish cost of a node is the sum of all shortest path distances to the other nodes, we use the probabilistic method to provide a new, structural characterization of equilibrium graphs. We show how to use this characterization in order to prove upper bounds on the diameter of equilibrium graphs in terms of the size of the largest $k$-vicinity (defined as the the set of vertices within distance $k$ from a vertex), for any $k \geq 1$ and in terms of the number of edges, thus settling positively a conjecture of Alon et al. in the cases of graphs of large $k$-vicinity size (including graphs of large maximum degree) and of graphs which are dense enough. Next, we present a new swap-based network creation game, in which selfish costs depend on the immediate neighborhood of each node; in particular, the profit of a node is defined as the sum of the degrees of its neighbors. We prove that, in contrast to the previous model, this network creation game admits an exact potential, and also that any equilibrium graph contains an induced star. The existence of the potential function is exploited in order to show that an equilibrium can be reached in expected polynomial time even in the case where nodes can only acquire limited knowledge concerning non-neighboring nodes.Comment: 11 pages, 4 figure

On a Bounded Budget Network Creation Game

We consider a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In our model, each link has unit price and each agent tries to minimize its cost, which is either its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two versions of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players' budgets is n-1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM versions, respectively. When each vertex has unit budget (i.e. can establish link to just one vertex), the diameter of any equilibrium graph in either version is Theta(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we show that if the budget of each player is at least k, then every equilibrium graph in the SUM version is k-connected or has diameter smaller than 4.Comment: 28 pages, 3 figures, preliminary version appeared in SPAA'1

Celebrity games

We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance ß as well as a link cost a. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than ß ) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if ß>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance ß introduced in Bilò et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft

Multilevel Network Games

We consider a multilevel network game, where nodes can improve their communication costs by connecting to a high-speed network. The $n$ nodes are connected by a static network and each node can decide individually to become a gateway to the high-speed network. The goal of a node $v$ is to minimize its private costs, i.e., the sum (SUM-game) or maximum (MAX-game) of communication distances from $v$ to all other nodes plus a fixed price $\alpha > 0$ if it decides to be a gateway. Between gateways the communication distance is $0$, and gateways also improve other nodes' distances by behaving as shortcuts. For the SUM-game, we show that for $\alpha \leq n-1$, the price of anarchy is $\Theta(n/\sqrt{\alpha})$ and in this range equilibria always exist. In range $\alpha \in (n-1,n(n-1))$ the price of anarchy is $\Theta(\sqrt{\alpha})$, and for $\alpha \geq n(n-1)$ it is constant. For the MAX-game, we show that the price of anarchy is either $\Theta(1 + n/\sqrt{\alpha})$, for $\alpha\geq 1$, or else $1$. Given a graph with girth of at least $4\alpha$, equilibria always exist. Concerning the dynamics, both the SUM-game and the MAX-game are not potential games. For the SUM-game, we even show that it is not weakly acyclic.Comment: An extended abstract of this paper has been accepted for publication in the proceedings of the 10th International Conference on Web and Internet Economics (WINE

Quality of Service in Network Creation Games

Network creation games model the creation and usage costs of networks formed by n selfish nodes. Each node v can buy a set of edges, each for a fixed price \alpha > 0. Its goal is to minimize its private costs, i.e., the sum (SUM-game, Fabrikant et al., PODC 2003) or maximum (MAX-game, Demaine et al., PODC 2007) of distances from $v$ to all other nodes plus the prices of the bought edges. The above papers show the existence of Nash equilibria as well as upper and lower bounds for the prices of anarchy and stability. In several subsequent papers, these bounds were improved for a wide range of prices \alpha. In this paper, we extend these models by incorporating quality-of-service aspects: Each edge cannot only be bought at a fixed quality (edge length one) for a fixed price \alpha. Instead, we assume that quality levels (i.e., edge lengths) are varying in a fixed interval [\beta,B], 0 < \beta <= B. A node now cannot only choose which edge to buy, but can also choose its quality x, for the price p(x), for a given price function p. For both games and all price functions, we show that Nash equilibria exist and that the price of stability is either constant or depends only on the interval size of available edge lengths. Our main results are bounds for the price of anarchy. In case of the SUM-game, we show that they are tight if price functions decrease sufficiently fast.Comment: An extended abstract of this paper has been accepted for publication in the proceedings of the 10th International Conference on Web and Internet Economics (WINE

The Max-Distance Network Creation Game on General Host Graphs

In this paper we study a generalization of the classic \emph{network creation game} in the scenario in which the $n$ players sit on a given arbitrary \emph{host graph}, which constrains the set of edges a player can activate at a cost of $\alpha \geq 0$ each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its \emph{maximum} distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of $\Omega (\sqrt{ n / (1+\alpha)})$ for any $\alpha = o(n)$. Notice that this implies a counter-intuitive lower bound of $\Omega(\sqrt{n})$ for very small values of $\alpha$ (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either $k$-regular (for any constant $k \geq 3$), or a 2-dimensional grid, the PoA is still $\Omega(1+\min\{\alpha, \frac{n}{\alpha}\})$, which is proven to be tight for $\alpha=\Omega(\sqrt{n})$. On the positive side, if $\alpha \geq n$, we show the PoA is $O(1)$. Finally, in the case in which the host graph is very sparse (i.e., $|E(H)|=n-1+k$, with $k=O(1)$), we prove that the PoA is $O(1)$, for any $\alpha$.Comment: 17 pages, 4 figure