Plurality Consensus in the Gossip Model

Abstract

We study Plurality Consensus in the GOSSIP Model over a network of n anonymous agents. Each agent supports an initial opinion or color. We assume that at the onset, the number of agents supporting the plurality color exceeds that of the agents supporting any other color by a sufficiently-large bias, though the initial plurality itself might be very far from absolute majority. The goal is to provide a protocol that, with high probability, brings the system into the configuration in which all agents support the (initial) plurality color.We consider the Undecided-State Dynamics, a wellknown protocol which uses just one more state (the undecided one) than those necessary to store colors.We show that the speed of convergence of this protocol depends on the initial color configuration as a whole, not just on the gap between the plurality and the second largest color community. This dependence is best captured by a novel notion we introduce, namely, the monochromatic distance md(c) which measures the distance of the initial color configurationc from the closest monochromatic one. In the complete graph, we prove that, for a wide range of the input parameters, this dynamics converges within O(md(c) log n) rounds. We prove that this upper bound is almost tight in the strong sense: Starting from any color configurationc, the convergence time is Ω(md(c)).Finally, we adapt the Undecided-State Dynamics to obtain a fast, random walk-based protocol for plurality consensus on regular expanders. This protocol converges in O(md(c) polylog(n)) rounds using only polylog(n) local memory. A key-ingredient to achieve the above bounds is a new analysis of the maximum node congestion that results from performing n parallel random walks on regular expanders.All our bounds hold with high probability.

Self-stabilizing repeated balls-into-bins

“…The above result strongly improves over the best previous bounds [12][13][14] and it is almost tight, since the classical lower bound Ω(log n/ log log n) on the maximum load (see, e.g., [11]) clearly applies also in our repeated setting. Our result further implies that, under the FIFO queueing policy, any ball performs Ω(t/ log n) steps of its individual random walk over any sequence of t = poly(n) rounds w.h.p., so the parallel cover time is O n log 2 n w.h.p.…”

“…In particular, in [13], a logarithmic bound is shown for the complete graph when m = O(n/ log n) random walks are performed over a logarithmic time interval, while a similar bound is also given for some families of almost-regular random graphs in [14]. A new analysis is given in [12] for regular graphs and time intervals of arbitrary length, yielding the bound O √ t . Finally, after the conference version of this paper [17], a probabilistic version of the Tetris process, where the number of new balls arriving at each round is a random variable with expectation λn, for some λ = λ(n) ∈ [0, 1], has been studied in [18].…”

“…The repeated balls-into-bins process was first considered in [13] (see there Lemma 2 in Subsection 3.1), [12] (see there Theorem 4.1 in Sect. 4), and [14] (see there Lemma 6 in Subsection 3.1), since it describes the process of performing parallel random walks in the (uniform) gossip model (also known as random phone-call model [15,16]) when every message can contain at most one token.…”

“…The previous approach in [12] relies on the fact that, in every round, the expected balance between the number of incoming and outgoing balls is always non-positive for every non-empty bin (notice that the expected number of incoming balls is always at most one). This may suggest viewing the process as a sort of parallel birth-death process [10].…”

“…This may suggest viewing the process as a sort of parallel birth-death process [10]. Using this approach and with some further arguments, one can (only) get the "standard-deviation" bound O( √ t) in [12]. Our new analysis proving Theorem 1 proceeds along three main steps.…”