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

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 α 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

On the Tree Conjecture for the Network Creation Game

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC\u2703] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees