388 research outputs found
Simple and Efficient Local Codes for Distributed Stable Network Construction
In this work, we study protocols so that populations of distributed processes
can construct networks. In order to highlight the basic principles of
distributed network construction we keep the model minimal in all respects. In
particular, we assume finite-state processes that all begin from the same
initial state and all execute the same protocol (i.e. the system is
homogeneous). Moreover, we assume pairwise interactions between the processes
that are scheduled by an adversary. The only constraint on the adversary
scheduler is that it must be fair. In order to allow processes to construct
networks, we let them activate and deactivate their pairwise connections. When
two processes interact, the protocol takes as input the states of the processes
and the state of the their connection and updates all of them. Initially all
connections are inactive and the goal is for the processes, after interacting
and activating/deactivating connections for a while, to end up with a desired
stable network. We give protocols (optimal in some cases) and lower bounds for
several basic network construction problems such as spanning line, spanning
ring, spanning star, and regular network. We provide proofs of correctness for
all of our protocols and analyze the expected time to convergence of most of
them under a uniform random scheduler that selects the next pair of interacting
processes uniformly at random from all such pairs. Finally, we prove several
universality results by presenting generic protocols that are capable of
simulating a Turing Machine (TM) and exploiting it in order to construct a
large class of networks.Comment: 43 pages, 7 figure
Moving in temporal graphs with very sparse random availability of edges
In this work we consider temporal graphs, i.e. graphs, each edge of which is
assigned a set of discrete time-labels drawn from a set of integers. The labels
of an edge indicate the discrete moments in time at which the edge is
available. We also consider temporal paths in a temporal graph, i.e. paths
whose edges are assigned a strictly increasing sequence of labels. Furthermore,
we assume the uniform case (UNI-CASE), in which every edge of a graph is
assigned exactly one time label from a set of integers and the time labels
assigned to the edges of the graph are chosen randomly and independently, with
the selection following the uniform distribution. We call uniform random
temporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving the
expected number of temporal paths of a given length in the uniform random
temporal clique. We define the term temporal distance of two vertices, which is
the arrival time, i.e. the time-label of the last edge, of the temporal path
that connects those vertices, which has the smallest arrival time amongst all
temporal paths that connect those vertices. We then propose and study two
statistical properties of temporal graphs. One is the maximum expected temporal
distance which is, as the term indicates, the maximum of all expected temporal
distances in the graph. The other one is the temporal diameter which, loosely
speaking, is the expectation of the maximum temporal distance in the graph. We
derive the maximum expected temporal distance of a uniform random temporal star
graph as well as an upper bound on both the maximum expected temporal distance
and the temporal diameter of the normalized version of the uniform random
temporal clique, in which the largest time-label available equals the number of
vertices. Finally, we provide an algorithm that solves an optimization problem
on a specific type of temporal (multi)graphs of two vertices.Comment: 30 page
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game
- …