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Distributed Computing

, Volume 30, Issue 4, pp 293–306 | Cite as

Simple dynamics for plurality consensus

  • Luca Becchetti
  • Andrea Clementi
  • Emanuele Natale
  • Francesco Pasquale
  • Riccardo Silvestri
  • Luca Trevisan
Article

Abstract

We study a plurality-consensus process in which each of n anonymous agents of a communication network initially supports a color chosen from the set [k]. Then, in every round, each agent can revise his color according to the colors 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 \(\mathcal {O}( \min \{ k, (n/\log n)^{1/3} \} \, \log n )\) with high probability, provided that \(s \geqslant c \sqrt{ \min \{ 2k, (n/\log n)^{1/3} \}\, n \log n}\). We then prove that our upper bound above is tight as long as \(k \leqslant (n/\log n)^{1/4}\). This fact implies an exponential time-gap between the plurality-consensus process and the median process (see Doerr et al. in Proceedings of the 23rd annual ACM symposium on parallelism in algorithms and architectures (SPAA’11), pp 149–158. ACM, 2011). 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.

Keywords

Plurality consensus Distributed randomized algorithms Markov chains Dynamics 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Sapienza Università di RomaRomeItaly
  2. 2.Max Planck Institute for InformaticsSaarbrückenGermany
  3. 3.Università Tor Vergata di RomaRomeItaly
  4. 4.RomeItaly
  5. 5.UC BerkeleyBerkeleyUSA

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