Network creation games: structure vs anarchy

Theory of Computing Systems

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'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of α 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 α and that for α ≥ n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price α and employ it to improve on the best known bounds for both conjectures. In particular we show that for α > 4n − 13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of α. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

Web and Internet Economics, 2019

We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other n − 1 players at a prefixed price α > 0. The player's goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is constant for all α. This conjecture has been confirmed for α = O(n 1−δ) with δ ≥ 1/ log n and for α > 4n − 13. The best known upper bound on the price of anarchy for the remaining range is 2 O(√ log n). We give new insights into the structure of the Nash equilibria for α > n and we enlarge the range of the parameter α for which the price of anarchy is constant. Specifically, we prove that for any small > 0, the price of anarchy is constant for α > n(1 +) by showing that any biconnected component of any non-trivial Nash equilibrium, if it exists, has at most a constant number of nodes.

ArXiv, 2020

We study the sum classic network creation game introduced by Fabrikant et al. in which n players conform a network buying links at individual price α. When studying this model we are mostly interested in Nash equilibria (ne) and the Price of Anarchy (PoA). It is conjectured that the PoA is constant for any α. Up until now, it has been proved constant PoA for the range α = O(n1−δ1) with δ1 > 0 a positive constant, upper bounding by a constant the diameter of any ne graph jointly with the fact that the diameter of any ne graph plus one unit is an upper bound for the PoA of the same graph. Also, it has been proved constant PoA for the range α > n(1 + δ2) with δ2 > 0 a positive constant, studying extensively the average degree of any biconnected component from equilibria. Our contribution consists in proving that ne graphs satisfy very restrictive topological properties generalising some properties proved in the literature and providing new insights that might help settling the...

ArXiv, 2018

We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price $\alpha > 0$ in order to get connected to the network formed by all the $n$ agents. In this setting, the reformulated tree conjecture states that for $\alpha > n$, every Nash equilibrium network is a tree. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. 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 $\alpha > n$. Up to now it has been proved that the \PoA is constant $(i)$ in the \emph{lower range}, for $\alpha = O(n^{1-\delta})$ with $\delta \geq \frac{1}{\log n}$ and $(ii)$ in the \emph{upper range}, for $\alpha > 4n-13$. In contrast, the best upper bound known for the price of anarchy for the remaining range is $2^{O(\sqrt{\log n})}$. I...

ArXiv, 2021

We study Nash equilibria in the network creation game of Fabrikant et al. [10]. In this game a vertex can buy an edge to another vertex for a cost of α, and the objective of each vertex is to minimize the sum of the costs of the edges it purchases plus the sum of the distances to every other vertex in the resultant network. A long-standing conjecture states that if α ≥ n then every Nash equilibrium in the game is a spanning tree. We prove the conjecture holds for any α > 3n− 3.

Arxiv preprint arXiv:1108.4115, 2011

We model the formation of networks as the result of a game where by players act selfishly to get the portfolio of links they desire most. The integration of player strategies into the network formation model is appropriate for organizational networks because in these smaller networks, dynamics are not random, but the result of intentional actions carried through by players maximizing their own objectives. This model is a better framework for the analysis of influences upon a network because it integrates the strategies of the players involved. We present an Integer Program that calculates the price of anarchy of this game by finding the worst stable graph and the best coordinated graph for this game. We simulate the formation of the network and calculated the simulated price of anarchy, which we find tends to be rather low.

2011

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 * A preliminary version of this paper appeared in 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 Θ(n) and Θ(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 Θ(1).

2011

We model the formation of networks as the result of a game where by players act selfishly to get the portfolio of links they desire most. The integration of player strategies into the network formation model is appropriate for organizational networks because in these smaller networks, dynamics are not random, but the result of intentional actions carried through by players maximizing their own objectives. This model is a better framework for the analysis of influences upon a network because it integrates the strategies of the players ...

Proceedings of the twenty-second annual symposium on Principles of distributed computing - PODC '03, 2003

We introduce a novel game that models the creation of Internet-like networks by selfish node-agents without central design or coordination. Nodes pay for the links that they establish, and benefit from short paths to all destinations. We study the Nash equilibria of this game, and prove results suggesting that the "price of anarchy" [4] in this context (the relative cost of the lack of coordination) may be modest. Several interesting extensions are suggested.

Eprint Arxiv Cs 0701071, 2007

Motivated by applications in peer-to-peer and overlay networks we define and study the \emph{Bounded Degree Network Formation} (BDNF) game. In an $(n,k)$-BDNF game, we are given $n$ nodes, a bound $k$ on the out-degree of each node, and a weight $w_{vu}$ for each ordered pair $(v,u)$ representing the traffic rate from node $v$ to node $u$. Each node $v$ uses up to $k$ directed links to connect to other nodes with an objective to minimize its average distance, using weights $w_{vu}$, to all other destinations. We study the existence of pure Nash equilibria for $(n,k)$-BDNF games. We show that if the weights are arbitrary, then a pure Nash wiring may not exist. Furthermore, it is NP-hard to determine whether a pure Nash wiring exists for a given $(n,k)$-BDNF instance. A major focus of this paper is on uniform $(n,k)$-BDNF games, in which all weights are 1. We describe how to construct a pure Nash equilibrium wiring given any $n$ and $k$, and establish that in all pure Nash wirings the cost of individual nodes cannot differ by more than a factor of nearly 2, whereas the diameter cannot exceed $O(\sqrt{n \log_k n})$. We also analyze best-response walks on the configuration space defined by the uniform game, and show that starting from any initial configuration, strong connectivity is reached within $\Theta(n^2)$ rounds. Convergence to a pure Nash equilibrium, however, is not guaranteed. We present simulation results that suggest that loop-free best-response walks always exist, but may not be polynomially bounded. We also study a special family of \emph{regular} wirings, the class of Abelian Cayley graphs, in which all nodes imitate the same wiring pattern, and show that if $n$ is sufficiently large no such regular wiring can be a pure Nash equilibrium.