Simple Dynamics for Plurality Consensus
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by
Luca Becchetti, Andrea Clementi, Emanuele Natale, Francesco Pasquale,
Riccardo Silvestri, Luca Trevisan
2015
Abstract
We study a Plurality-Consensus process in which each of n anonymous
agents of a communication network initially supports an opinion (a color chosen
from a finite set [k]). Then, in every (synchronous) round, each agent can
revise his color according to the opinions currently held by a random sample of
his neighbors. It is assumed that the initial color configuration exhibits a
sufficiently large bias s towards a fixed plurality color, that is,
the number of nodes supporting the plurality color exceeds the number of nodes
supporting any other color by s additional nodes. The goal is having the
process to converge to the stable configuration in which all nodes
support the initial plurality. We consider a basic model in which the network
is a clique and the update rule (called here the 3-majority dynamics) of
the process is the following: each agent looks at the colors of three random
neighbors and then applies the majority rule (breaking ties uniformly).
We prove that the process converges in time O( { k, (n/
n)^1/3} n ) with high probability, provided that s ≥ c
√({ 2k, (n/ n)^1/3} n n).
We then prove that our upper bound above is tight as long as k ≤
(n/ n)^1/4. This fact implies an exponential time-gap between the
plurality-consensus process and the median process studied by Doerr et
al. in [ACM SPAA'11].
A natural question is whether looking at more (than three) random neighbors
can significantly speed up the process. We provide a negative answer to this
question: In particular, we show that samples of polylogarithmic size can speed
up the process by a polylogarithmic factor only.
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