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.
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).
We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Ω( √ 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( √ 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.
In recent years, a lot of research has been conducted on network design problems, because of their importance in computer science and operations research [7,16,21]. The aim in this line of research is usually to build a minimum cost network that satisfies certain properties, and the network structure is usually determined by a central authority. However, this is in contrast to many real world situations such as social networks, client-server systems and peer-to-peer networks, where network structures are determined in a distributed manner by selfish agents [12,13,22]. The formation of these networks can be formulated as a game, which is usually called a network creation game. In network creation games, as in any other game, there are selfish players that interact with each other. Each player has its own objective, and attempts to minimize the cost it incurs in the network, regardless of how its actions affect other agents. The players are placed at the nodes of the network graph, and can create links to other nodes with certain restrictions, e.g. there could be an upper bound for the number of links a player constructs. The utility functions of the players should be defined properly to be consistent with their natural interests, e.g. minimizing the cost to communicate with other players.
In network creation games, players interact with each other by adding and removing links between themselves. Many variants of these games arise by defining different utility functions and possible transitions between strategies of each player. Some authors have studied undirected graphs while others have considered directed graphs. For undirected graphs there is an issue of “ownership”: when there exists a link between two nodes, but just one of the nodes wants to keep it, is it removed from the network? For directed graphs, the question is whether both endpoints of a link can use it to communicate. In some models creating a link incurs a cost to the player, i.e. the number of created links appears in utility functions, while in other models restrictions for creating links appear in the set of available strategies for players.
Although the creation of a network in such a game is a dynamic process, the structure of the resulting network (if it does converge to a stable structure) provides valuable information about the effectiveness of the game rules. Thus, existence and structure of stable networks has been widely studied. Jackson and Wolinsky [15] defined a network to be pairwise stable if, roughly speaking, none of the nodes are willing to delete an incident link and no pair of non-adjacent nodes are willing to build a link between themselves. Fabrikant et al. [11] considered Nash equilibria of the game as its stable states. A Nash equilibrium, which is a well known concept in game theory, is a state of a game in which no player can increase her utility by changing her strategy, assuming the strategies of other players are kept unchanged. The main difference between these two concepts is that, when considering pairwise stability, we are thinking of the players cooperating with each other, whereas we think of non-cooperating players when we consider Nash equilibria.
The efficiency of the network formed by a game is measured by different factors rather than the player utilities. We are mostly interested in measuring a global parameter, for instance the diameter (the largest distance between any pair of nodes), and the vertex connectivity (the minimum number of nodes whose removal disconnects the network) of the network ar
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