217,314 research outputs found
Efficient Local Search in Coordination Games on Graphs
We study strategic games on weighted directed graphs, where the payoff of a
player is defined as the sum of the weights on the edges from players who chose
the same strategy augmented by a fixed non-negative bonus for picking a given
strategy. These games capture the idea of coordination in the absence of
globally common strategies. Prior work shows that the problem of determining
the existence of a pure Nash equilibrium for these games is NP-complete already
for graphs with all weights equal to one and no bonuses. However, for several
classes of graphs (e.g. DAGs and cliques) pure Nash equilibria or even strong
equilibria always exist and can be found by simply following a particular
improvement or coalition-improvement path, respectively. In this paper we
identify several natural classes of graphs for which a finite improvement or
coalition-improvement path of polynomial length always exists, and, as a
consequence, a Nash equilibrium or strong equilibrium in them can be found in
polynomial time. We also argue that these results are optimal in the sense that
in natural generalisations of these classes of graphs, a pure Nash equilibrium
may not even exist.Comment: Extended version of a paper accepted to IJCAI1
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
Equilibrium statistical mechanics on correlated random graphs
Biological and social networks have recently attracted enormous attention
between physicists. Among several, two main aspects may be stressed: A non
trivial topology of the graph describing the mutual interactions between agents
exists and/or, typically, such interactions are essentially (weighted)
imitative. Despite such aspects are widely accepted and empirically confirmed,
the schemes currently exploited in order to generate the expected topology are
based on a-priori assumptions and in most cases still implement constant
intensities for links. Here we propose a simple shift in the definition of
patterns in an Hopfield model to convert frustration into dilution: By varying
the bias of the pattern distribution, the network topology -which is generated
by the reciprocal affinities among agents - crosses various well known regimes
(fully connected, linearly diverging connectivity, extreme dilution scenario,
no network), coupled with small world properties, which, in this context, are
emergent and no longer imposed a-priori. The model is investigated at first
focusing on these topological properties of the emergent network, then its
thermodynamics is analytically solved (at a replica symmetric level) by
extending the double stochastic stability technique, and presented together
with its fluctuation theory for a picture of criticality. At least at
equilibrium, dilution simply decreases the strength of the coupling felt by the
spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main
difference with respect to previous investigations and a naive picture is that
within our approach replicas do not appear: instead of (multi)-overlaps as
order parameters, we introduce a class of magnetizations on all the possible
sub-graphs belonging to the main one investigated: As a consequence, for these
objects a closure for a self-consistent relation is achieved.Comment: 30 pages, 4 figure
A Characterization of Undirected Graphs Admitting Optimal Cost Shares
In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing
protocols for network design with the objective to implement a low-cost Steiner
forest as a Nash equilibrium of an induced cost-sharing game. One of the most
intriguing open problems to date is to understand the power of budget-balanced
and separable cost sharing protocols in order to induce low-cost Steiner
forests. In this work, we focus on undirected networks and analyze topological
properties of the underlying graph so that an optimal Steiner forest can be
implemented as a Nash equilibrium (by some separable cost sharing protocol)
independent of the edge costs. We term a graph efficient if the above stated
property holds. As our main result, we give a complete characterization of
efficient undirected graphs for two-player network design games: an undirected
graph is efficient if and only if it does not contain (at least) one out of few
forbidden subgraphs. Our characterization implies that several graph classes
are efficient: generalized series-parallel graphs, fan and wheel graphs and
graphs with small cycles.Comment: 60 pages, 69 figures, OR 2017 Berlin, WINE 2017 Bangalor
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
- …