19,485 research outputs found
Random walks and voting theory
Voters' preferences depend on the available information. Following Case-Based Decision Theory, we assume that this information is processed additively. We prove that the collective preferences deduced from the individual ones through majority vote cannot be arbitrary, as soon as a winning quota is required. The proof is based on a new result on random walks.voting theory; quotas; random walks
The Power of Two Choices in Distributed Voting
Distributed voting is a fundamental topic in distributed computing. In pull
voting, in each step every vertex chooses a neighbour uniformly at random, and
adopts its opinion. The voting is completed when all vertices hold the same
opinion. On many graph classes including regular graphs, pull voting requires
expected steps to complete, even if initially there are only two
distinct opinions.
In this paper we consider a related process which we call two-sample voting:
every vertex chooses two random neighbours in each step. If the opinions of
these neighbours coincide, then the vertex revises its opinion according to the
chosen sample. Otherwise, it keeps its own opinion. We consider the performance
of this process in the case where two different opinions reside on vertices of
some (arbitrary) sets and , respectively. Here, is the
number of vertices of the graph.
We show that there is a constant such that if the initial imbalance
between the two opinions is ?, then with high probability two sample voting completes in a random
regular graph in steps and the initial majority opinion wins. We
also show the same performance for any regular graph, if where is the second largest eigenvalue of the transition
matrix. In the graphs we consider, standard pull voting requires
steps, and the minority can still win with probability .Comment: 22 page
Fast plurality consensus in regular expanders
Pull voting is a classic method to reach consensus among vertices with
differing opinions in a distributed network: each vertex at each step takes on
the opinion of a random neighbour. This method, however, suffers from two
drawbacks. Even if there are only two opposing opinions, the time taken for a
single opinion to emerge can be slow and the final opinion is not necessarily
the initially held majority.
We refer to a protocol where 2 neighbours are contacted at each step as a
2-sample voting protocol. In the two-sample protocol a vertex updates its
opinion only if both sampled opinions are the same. Not much was known about
the performance of two-sample voting on general expanders in the case of three
or more opinions. In this paper we show that the following performance can be
achieved on a -regular expander using two-sample voting. We suppose there
are opinions, and that the initial size of the largest and second
largest opinions is respectively.
We prove that, if ,
where is the absolute second eigenvalue of matrix and
is a suitable constant, then the largest opinion wins in steps with high probability.
For almost all -regular graphs, we have for some
constant . This means that as increases we can separate an opinion
whose majority is , whereas majority is required for
constant.
This work generalizes the results of Becchetti et. al (SPAA 2014) for the
complete graph
Phase Transitions of Best-of-Two and Best-of-Three on Stochastic Block Models
This paper is concerned with voting processes on graphs where each vertex
holds one of two different opinions. In particular, we study the
\emph{Best-of-two} and the \emph{Best-of-three}. Here at each synchronous and
discrete time step, each vertex updates its opinion to match the majority among
the opinions of two random neighbors and itself (the Best-of-two) or the
opinions of three random neighbors (the Best-of-three). Previous studies have
explored these processes on complete graphs and expander graphs, but we
understand significantly less about their properties on graphs with more
complicated structures.
In this paper, we study the Best-of-two and the Best-of-three on the
stochastic block model , which is a random graph consisting of two
distinct Erd\H{o}s-R\'enyi graphs joined by random edges with density
. We obtain two main results. First, if and
is a constant, we show that there is a phase transition in with
threshold (specifically, for the Best-of-two, and
for the Best-of-three). If , the process reaches consensus
within steps for any initial opinion
configuration with a bias of . By contrast, if , then there
exists an initial opinion configuration with a bias of from which
the process requires at least steps to reach consensus. Second,
if is a constant and , we show that, for any initial opinion
configuration, the process reaches consensus within steps. To the
best of our knowledge, this is the first result concerning multiple-choice
voting for arbitrary initial opinion configurations on non-complete graphs
An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs
We analyze a class of distributed quantized consensus algorithms for
arbitrary static networks. In the initial setting, each node in the network has
an integer value. Nodes exchange their current estimate of the mean value in
the network, and then update their estimation by communicating with their
neighbors in a limited capacity channel in an asynchronous clock setting.
Eventually, all nodes reach consensus with quantized precision. We analyze the
expected convergence time for the general quantized consensus algorithm
proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric
networks, random walks, and couplings of Markov chains to derive an upper bound for the expected convergence time on an arbitrary graph of size
, improving on the state of art bound of for quantized consensus
algorithms. Our result is not dependent on graph topology. Example of complete
graphs is given to show how to extend the analysis to graphs of given topology.Comment: to appear in IEEE Trans. on Automatic Control, January, 2015. arXiv
admin note: substantial text overlap with arXiv:1208.078
Bounds on the Voter Model in Dynamic Networks
In the voter model, each node of a graph has an opinion, and in every round
each node chooses independently a random neighbour and adopts its opinion. We
are interested in the consensus time, which is the first point in time where
all nodes have the same opinion. We consider dynamic graphs in which the edges
are rewired in every round (by an adversary) giving rise to the graph sequence
, where we assume that has conductance at least
. We assume that the degrees of nodes don't change over time as one can
show that the consensus time can become super-exponential otherwise. In the
case of a sequence of -regular graphs, we obtain asymptotically tight
results. Even for some static graphs, such as the cycle, our results improve
the state of the art. Here we show that the expected number of rounds until all
nodes have the same opinion is bounded by , for any
graph with edges, conductance , and degrees at least . In
addition, we consider a biased dynamic voter model, where each opinion is
associated with a probability , and when a node chooses a neighbour with
that opinion, it adopts opinion with probability (otherwise the node
keeps its current opinion). We show for any regular dynamic graph, that if
there is an difference between the highest and second highest
opinion probabilities, and at least nodes have initially the
opinion with the highest probability, then all nodes adopt w.h.p. that opinion.
We obtain a bound on the convergences time, which becomes for
static graphs
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