Simple Dynamics for Plurality Consensus

We study a Plurality 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. The goal (of the agents) is to let the process converge to the stable configuration where all nodes support the plurality color. It is assumed that the initial color configuration has a sufficiently large bias s, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by an additive value s. 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 at random). We prove a tight bound on the convergence time which grows as Theta(k log n) for a wide range of parameters k and n. This linear-in-k dependence implies an exponential time-gap between the plurality consensus process and the median process studied in [7]. 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.