1,085 research outputs found
Consensus Over Martingale Graph Processes
In this paper, we consider a consensus seeking process based on repeated averaging in a randomly changing network. The underlying graph of such a network at each time is generated by a martingale random process. We prove that consensus is reached almost surely if and only if the expected graph of the network contains a directed spanning tree. We then provide an example of a consensus seeking process based on local averaging of opinions in a dynamic model of social network formation which is a martingale. At each time step, individual agents randomly choose some other agents to interact with according to some arbitrary probabilities. The interaction is one-sided and results in the agent averaging her opinion with those of her randomly chosen neighbors based on the weights she assigns to them. Once an agent chooses a neighbor, the weights are updated in such a way that the expected values of the weights are preserved. We show that agents reach consensus in this random dynamical network almost surely. Finally, we demonstrate that a Polya Urn process is a martingale process, and our prior results in [1] is a special case of the model proposed in this paper
Consensus time in a voter model with concealed and publicly expressed opinions
The voter model is a simple agent-based model to mimic opinion dynamics in
social networks: a randomly chosen agent adopts the opinion of a randomly
chosen neighbour. This process is repeated until a consensus emerges. Although
the basic voter model is theoretically intriguing, it misses an important
feature of real opinion dynamics: it does not distinguish between an agent's
publicly expressed opinion and her inner conviction. A person may not feel
comfortable declaring her conviction if her social circle appears to hold an
opposing view. Here we introduce the Concealed Voter Model where we add a
second, concealed layer of opinions to the public layer. If an agent's public
and concealed opinions disagree, she can reconcile them by either publicly
disclosing her previously secret point of view or by accepting her public
opinion as inner conviction. We study a complete graph of agents who can choose
from two opinions. We define a martingale that determines the probability
of all agents eventually agreeing on a particular opinion. By analyzing the
evolution of in the limit of a large number of agents, we derive the
leading-order terms for the mean and standard deviation of the consensus time
(i.e. the time needed until all opinions are identical). We thereby give a
precise prediction by how much concealed opinions slow down a consensus.Comment: 21 pages, 6 figures, to appear in J. Stat. Mech. Theory Ex
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
Reaching consensus on a connected graph
We study a simple random process in which vertices of a connected graph reach
consensus through pairwise interactions. We compute outcome probabilities,
which do not depend on the graph structure, and consider the expected time
until a consensus is reached. In some cases we are able to show that this is
minimised by . We prove an upper bound for the case and give a
family of graphs which asymptotically achieve this bound. In order to obtain
the mean of the waiting time we also study a gambler's ruin process with
delays. We give the mean absorption time and prove that it monotonically
increases with for symmetric delays
On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols
In this work we focus on a natural class of population protocols whose
dynamics are modelled by the discrete version of Lotka-Volterra equations. In
such protocols, when an agent of type (species) interacts with an agent
of type (species) with as the initiator, then 's type becomes
with probability . In such an interaction, we think of as the
predator, as the prey, and the type of the prey is either converted to that
of the predator or stays as is. Such protocols capture the dynamics of some
opinion spreading models and generalize the well-known Rock-Paper-Scissors
discrete dynamics. We consider the pairwise interactions among agents that are
scheduled uniformly at random. We start by considering the convergence time and
show that any Lotka-Volterra-type protocol on an -agent population converges
to some absorbing state in time polynomial in , w.h.p., when any pair of
agents is allowed to interact. By contrast, when the interaction graph is a
star, even the Rock-Paper-Scissors protocol requires exponential time to
converge. We then study threshold effects exhibited by Lotka-Volterra-type
protocols with 3 and more species under interactions between any pair of
agents. We start by presenting a simple 4-type protocol in which the
probability difference of reaching the two possible absorbing states is
strongly amplified by the ratio of the initial populations of the two other
types, which are transient, but "control" convergence. We then prove that the
Rock-Paper-Scissors protocol reaches each of its three possible absorbing
states with almost equal probability, starting from any configuration
satisfying some sub-linear lower bound on the initial size of each species.
That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a
distributed system. Some of our techniques may be of independent value
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