Network Creation Games: Think Global - Act Local

We investigate a non-cooperative game-theoretic model for the formation of communication networks by selfish agents. Each agent aims for a central position at minimum cost for creating edges. In particular, the general model (Fabrikant et al., PODC'03) became popular for studying the structure of the Internet or social networks. Despite its significance, locality in this game was first studied only recently (Bil\`o et al., SPAA'14), where a worst case locality model was presented, which came with a high efficiency loss in terms of quality of equilibria. Our main contribution is a new and more optimistic view on locality: agents are limited in their knowledge and actions to their local view ranges, but can probe different strategies and finally choose the best. We study the influence of our locality notion on the hardness of computing best responses, convergence to equilibria, and quality of equilibria. Moreover, we compare the strength of local versus non-local strategy-changes. Our results address the gap between the original model and the worst case locality variant. On the bright side, our efficiency results are in line with observations from the original model, yet we have a non-constant lower bound on the price of anarchy.Comment: An extended abstract of this paper has been accepted for publication in the proceedings of the 40th International Conference on Mathematical Foundations on Computer Scienc

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

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

Selfish Network Creation with Non-Uniform Edge Cost

Network creation games investigate complex networks from a game-theoretic point of view. Based on the original model by Fabrikant et al. [PODC'03] many variants have been introduced. However, almost all versions have the drawback that edges are treated uniformly, i.e. every edge has the same cost and that this common parameter heavily influences the outcomes and the analysis of these games. We propose and analyze simple and natural parameter-free network creation games with non-uniform edge cost. Our models are inspired by social networks where the cost of forming a link is proportional to the popularity of the targeted node. Besides results on the complexity of computing a best response and on various properties of the sequential versions, we show that the most general version of our model has constant Price of Anarchy. To the best of our knowledge, this is the first proof of a constant Price of Anarchy for any network creation game.Comment: To appear at SAGT'1

The Price of Anarchy in Cooperative Network Creation Games

In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost \alpha. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation game. First, we study extensively a natural version of the game, called the cooperative model, where nodes can collaborate and share the cost of creating any edge in the host graph. We prove the first nontrivial bounds in this model, establishing that the price of anarchy is polylogarithmic in n for all values of &#945; in complete host graphs. This bound is the first result of this type for any version of the network creation game; most previous general upper bounds are polynomial in n. Interestingly, we also show that equilibrium graphs have polylogarithmic diameter for the most natural range of \alpha (at most n polylg n). Second, we study the impact of the natural assumption that the host graph is a general graph, not necessarily complete. This model is a simple example of nonuniform creation costs among the edges (effectively allowing weights of \alpha and \infty). We prove the first assemblage of upper and lower bounds for this context, stablishing nontrivial tight bounds for many ranges of \alpha, for both the unilateral and cooperative versions of network creation. In particular, we establish polynomial lower bounds for both versions and many ranges of \alpha, even for this simple nonuniform cost model, which sharply contrasts the conjectured constant bounds for these games in complete (uniform) graphs

Network creation games: structure vs anarchy

We study Nash equilibria and the price of anarchy in the classical model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price a > 0 in order to get connected to the network formed by all the n agents. In this setting, the reformulated tree conjecture states that for a > n, every Nash equilibrium network is a tree. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of anarchy would be constant for a > n. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. Up to now the last conjecture has been proven in (i) the lower range, for a = O(n1-o¿) with o¿ = 1 and (ii) in the upper range, for a > 65n. In ¿log n contrast, the best upper bound known for the price of anarchy for the remaining range is 2O(vlog n). In this paper we give new insights into the structure of the Nash equilibria for different ranges of a and we enlarge the range for which the price of anarchy is constant. Regarding the upper range, we prove that every Nash equilibrium is a tree for a > 17n and that the price of anarchy is constant even for a > 9n. In the lower range, we show that any Nash equilibrium for a 4, induces an o¿-distance-almost- uniform graph.Postprint (published version