33,564 research outputs found
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
Inequality and Network Formation Games
This paper addresses the matter of inequality in network formation games. We
employ a quantity that we are calling the Nash Inequality Ratio (NIR), defined
as the maximal ratio between the highest and lowest costs incurred to
individual agents in a Nash equilibrium strategy, to characterize the extent to
which inequality is possible in equilibrium. We give tight upper bounds on the
NIR for the network formation games of Fabrikant et al. (PODC '03) and Ehsani
et al. (SPAA '11). With respect to the relationship between equality and social
efficiency, we show that, contrary to common expectations, efficiency does not
necessarily come at the expense of increased inequality.Comment: 27 pages. 4 figures. Accepted to Internet Mathematics (2014
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 players sit on a given arbitrary
\emph{host graph}, which constrains the set of edges a player can activate at a
cost of 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
for any . Notice that this implies a counter-intuitive lower
bound of for very small values of (i.e., edges can
be activated almost for free). Then, we show that when the host graph is
restricted to be either -regular (for any constant ), or a
2-dimensional grid, the PoA is still , which is proven to be tight for
. On the positive side, if , we show
the PoA is . Finally, in the case in which the host graph is very sparse
(i.e., , with ), we prove that the PoA is , for any
.Comment: 17 pages, 4 figure
Self-Organizing Flows in Social Networks
Social networks offer users new means of accessing information, essentially
relying on "social filtering", i.e. propagation and filtering of information by
social contacts. The sheer amount of data flowing in these networks, combined
with the limited budget of attention of each user, makes it difficult to ensure
that social filtering brings relevant content to the interested users. Our
motivation in this paper is to measure to what extent self-organization of the
social network results in efficient social filtering. To this end we introduce
flow games, a simple abstraction that models network formation under selfish
user dynamics, featuring user-specific interests and budget of attention. In
the context of homogeneous user interests, we show that selfish dynamics
converge to a stable network structure (namely a pure Nash equilibrium) with
close-to-optimal information dissemination. We show in contrast, for the more
realistic case of heterogeneous interests, that convergence, if it occurs, may
lead to information dissemination that can be arbitrarily inefficient, as
captured by an unbounded "price of anarchy". Nevertheless the situation differs
when users' interests exhibit a particular structure, captured by a metric
space with low doubling dimension. In that case, natural autonomous dynamics
converge to a stable configuration. Moreover, users obtain all the information
of interest to them in the corresponding dissemination, provided their budget
of attention is logarithmic in the size of their interest set
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 -vicinity (defined as
the the set of vertices within distance from a vertex), for any
and in terms of the number of edges, thus settling positively a conjecture of
Alon et al. in the cases of graphs of large -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
Tree Nash Equilibria in the Network Creation Game
In the network creation game with n vertices, every vertex (a player) buys a
set of adjacent edges, each at a fixed amount {\alpha} > 0. It has been
conjectured that for {\alpha} >= n, every Nash equilibrium is a tree, and has
been confirmed for every {\alpha} >= 273n. We improve upon this bound and show
that this is true for every {\alpha} >= 65n. To show this, we provide new and
improved results on the local structure of Nash equilibria. Technically, we
show that if there is a cycle in a Nash equilibrium, then {\alpha} < 65n.
Proving this, we only consider relatively simple strategy changes of the
players involved in the cycle. We further show that this simple approach cannot
be used to show the desired upper bound {\alpha} < n (for which a cycle may
exist), but conjecture that a slightly worse bound {\alpha} < 1.3n can be
achieved with this approach. Towards this conjecture, we show that if a Nash
equilibrium has a cycle of length at most 10, then indeed {\alpha} < 1.3n. We
further provide experimental evidence suggesting that when the girth of a Nash
equilibrium is increasing, the upper bound on {\alpha} obtained by the simple
strategy changes is not increasing. To the end, we investigate the approach for
a coalitional variant of Nash equilibrium, where coalitions of two players
cannot collectively improve, and show that if {\alpha} >= 41n, then every such
Nash equilibrium is a tree
The Price of Anarchy for Network Formation in an Adversary Model
We study network formation with n players and link cost \alpha > 0. After the
network is built, an adversary randomly deletes one link according to a certain
probability distribution. Cost for player v incorporates the expected number of
players to which v will become disconnected. We show existence of equilibria
and a price of stability of 1+o(1) under moderate assumptions on the adversary
and n \geq 9.
As the main result, we prove bounds on the price of anarchy for two special
adversaries: one removes a link chosen uniformly at random, while the other
removes a link that causes a maximum number of player pairs to be separated.
For unilateral link formation we show a bound of O(1) on the price of anarchy
for both adversaries, the constant being bounded by 10+o(1) and 8+o(1),
respectively. For bilateral link formation we show O(1+\sqrt{n/\alpha}) for one
adversary (if \alpha > 1/2), and \Theta(n) for the other (if \alpha > 2
considered constant and n \geq 9). The latter is the worst that can happen for
any adversary in this model (if \alpha = \Omega(1)). This points out
substantial differences between unilateral and bilateral link formation
- …