Simple Dynamics for Majority Consensus

We study a "Majority Consensus" process in which each of $n$ anonymous agents of a communication network supports an initial opinion (a color chosen from a finite set $[k]$) and, at every time step, he can revise his color according to a random sample of neighbors. It is assumed that the initial color configuration has a sufficiently large bias $s$ towards a fixed majority color, that is, the number of nodes supporting the majority color exceeds the number of nodes supporting any other color by an additive factor $s$. The goal (of the agents) is to let the process converge to the stable configuration where all nodes support the majority color. 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 that 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 $O(\min\{k, (n/\log n)^{1/3}\} \log n)$ with high probability, provided that $s \geqslant c \sqrt{\min\{2k, (n/\log n)^{1/3}\} n\log n}$. Departing significantly from the previous analysis, our proof technique also yields a $\polylog(n)$ bound on the convergence time whenever the initial number of nodes supporting the majority color is larger than $n/\polylog(n)$ and $s \geqslant \sqrt{n\polylog(n)}$, no matter how large $k$ is. 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 majority-consensus process and the "median" process studied in [Doerr et al., SPAA'11]. A natural question is whether looking at more (than three) random neighbors might 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.