209 research outputs found
Robustness of large-scale stochastic matrices to localized perturbations
Upper bounds are derived on the total variation distance between the
invariant distributions of two stochastic matrices differing on a subset W of
rows. Such bounds depend on three parameters: the mixing time and the minimal
expected hitting time on W for the Markov chain associated to one of the
matrices; and the escape time from W for the Markov chain associated to the
other matrix. These results, obtained through coupling techniques, prove
particularly useful in scenarios where W is a small subset of the state space,
even if the difference between the two matrices is not small in any norm.
Several applications to large-scale network problems are discussed, including
robustness of Google's PageRank algorithm, distributed averaging and consensus
algorithms, and interacting particle systems.Comment: 12 pages, 4 figure
Scaling limits for continuous opinion dynamics systems
Scaling limits are analyzed for stochastic continuous opinion dynamics
systems, also known as gossip models. In such models, agents update their
vector-valued opinion to a convex combination (possibly agent- and
opinion-dependent) of their current value and that of another observed agent.
It is shown that, in the limit of large agent population size, the empirical
opinion density concentrates, at an exponential probability rate, around the
solution of a probability-measure-valued ordinary differential equation
describing the system's mean-field dynamics. Properties of the associated
initial value problem are studied. The asymptotic behavior of the solution is
analyzed for bounded-confidence opinion dynamics, and in the presence of an
heterogeneous influential environment.Comment: Published at http://dx.doi.org/10.1214/10-AAP739 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
From local averaging to emergent global behaviors: the fundamental role of network interconnections
Distributed averaging is one of the simplest and most studied network
dynamics. Its applications range from cooperative inference in sensor networks,
to robot formation, to opinion dynamics. A number of fundamental results and
examples scattered through the literature are gathered here and originally
presented, emphasizing the deep interplay between the network interconnection
structure and the emergent global behavior.Comment: 10 page
The asymptotical error of broadcast gossip averaging algorithms
In problems of estimation and control which involve a network, efficient
distributed computation of averages is a key issue. This paper presents
theoretical and simulation results about the accumulation of errors during the
computation of averages by means of iterative "broadcast gossip" algorithms.
Using martingale theory, we prove that the expectation of the accumulated error
can be bounded from above by a quantity which only depends on the mixing
parameter of the algorithm and on few properties of the network: its size, its
maximum degree and its spectral gap. Both analytical results and computer
simulations show that in several network topologies of applicative interest the
accumulated error goes to zero as the size of the network grows large.Comment: 10 pages, 3 figures. Based on a draft submitted to IFACWC201
The robustness of democratic consensus
In linear models of consensus dynamics, the state of the various agents
converges to a value which is a convex combination of the agents' initial
states. We call it democratic if in the large scale limit (number of agents
going to infinity) the vector of convex weights converges to 0 uniformly.
Democracy is a relevant property which naturally shows up when we deal with
opinion dynamic models and cooperative algorithms such as consensus over a
network: it says that each agent's measure/opinion is going to play a
negligeable role in the asymptotic behavior of the global system. It can be
seen as a relaxation of average consensus, where all agents have exactly the
same weight in the final value, which becomes negligible for a large number of
agents.Comment: 13 pages, 2 fig
A game theoretic approach to a peer-to-peer cloud storage model
Classical cloud storage based on external data providers has been recognized
to suffer from a number of drawbacks. This is due to its inherent centralized
architecture which makes it vulnerable to external attacks, malware, technical
failures, as well to the large premium charged for business purposes. In this
paper, we propose an alternative distributed peer-to-peer cloud storage model
which is based on the observation that the users themselves often have
available storage capabilities to be offered in principle to other users. Our
set-up is that of a network of users connected through a graph, each of them
being at the same time a source of data to be stored externally and a possible
storage resource. We cast the peer-to-peer storage model to a Potential Game
and we propose an original decentralized algorithm which makes units interact,
cooperate, and store a complete back up of their data on their connected
neighbors. We present theoretical results on the algorithm as well a good
number of simulations which validate our approach.Comment: 10 page
On imitation dynamics in potential population games
Imitation dynamics for population games are studied and their asymptotic
properties analyzed. In the considered class of imitation dynamics - that
encompass the replicator equation as well as other models previously considered
in evolutionary biology - players have no global information about the game
structure, and all they know is their own current utility and the one of fellow
players contacted through pairwise interactions. For potential population
games, global asymptotic stability of the set of Nash equilibria of the
sub-game restricted to the support of the initial population configuration is
proved. These results strengthen (from local to global asymptotic stability)
existing ones and generalize them to a broader class of dynamics. The developed
techniques highlight a certain structure of the problem and suggest possible
generalizations from the fully mixed population case to imitation dynamics
whereby agents interact on complex communication networks.Comment: 7 pages, 3 figures. Accepted at CDC 201
Finite-time influence systems and the Wisdom of Crowd effect
Recent contributions have studied how an influence system may affect the
wisdom of crowd phenomenon. In the so-called naive learning setting, a crowd of
individuals holds opinions that are statistically independent estimates of an
unknown parameter; the crowd is wise when the average opinion converges to the
true parameter in the limit of infinitely many individuals. Unfortunately, even
starting from wise initial opinions, a crowd subject to certain influence
systems may lose its wisdom. It is of great interest to characterize when an
influence system preserves the crowd wisdom effect. In this paper we introduce
and characterize numerous wisdom preservation properties of the basic
French-DeGroot influence system model. Instead of requiring complete
convergence to consensus as in the previous naive learning model by Golub and
Jackson, we study finite-time executions of the French-DeGroot influence
process and establish in this novel context the notion of prominent families
(as a group of individuals with outsize influence). Surprisingly, finite-time
wisdom preservation of the influence system is strictly distinct from its
infinite-time version. We provide a comprehensive treatment of various
finite-time wisdom preservation notions, counterexamples to meaningful
conjectures, and a complete characterization of equal-neighbor influence
systems
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