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 biass 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(min{k,(n/logn)1/3}logn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}( \min \{ k, (n/\log n)^{1/3} \} \, \log n )$$\end{document} with high probability, provided that s⩾cmin{2k,(n/logn)1/3}nlogn\documentclass[12pt]{minimal}
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\begin{document}$$s \geqslant c \sqrt{ \min \{ 2k, (n/\log n)^{1/3} \}\, n \log n}$$\end{document}. We then prove that our upper bound above is tight as long as k⩽(n/logn)1/4\documentclass[12pt]{minimal}
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\begin{document}$$k \leqslant (n/\log n)^{1/4}$$\end{document}. 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.
