Simple dynamics for plurality consensus

Becchetti, Luca; Clementi, Andrea E. F.; Natale, Emanuele; Pasquale, Francesco; Silvestri, Riccardo; Trevisan, Luca

doi:10.1007/s00446-016-0289-4

Cited by 44 publications

(93 citation statements)

References 24 publications

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“…A similar result has been obtained for another elementary protocol, so-called 3-majority dynamics, in which, at each round, each node samples the opinion of three random nodes, and adopts the most frequent opinion among these three [10]. The 3-majority dynamics has also been shown to be fault-tolerant against an adversary that can change up to O( √ n) agents at each round [10,20]. Other work has analyzed the undecided-state dynamics in asynchronous models with a constant number of opinions [21,31,37], and the h-majority dynamics (or slight variations of it) on different graph classes in the uniform push model [1,18].…”

Section: Other Related Worksupporting

confidence: 72%

“…Still in the uniform push model, the undecided state dynamic has been analyzed in the case of an arbitrarily large number of opinions, which may even be a function of the number of agents in the system [9]. A similar result has been obtained for another elementary protocol, so-called 3-majority dynamics, in which, at each round, each node samples the opinion of three random nodes, and adopts the most frequent opinion among these three [10]. The 3-majority dynamics has also been shown to be fault-tolerant against an adversary that can change up to O( √ n) agents at each round [10,20].…”

Section: Other Related Workmentioning

confidence: 81%

“…Which choice is better depends on the biological setting that is being modeled: if the communication between the agents of the system is an auditory or tactile signal, it could be more realistic to assume that simultaneous messages to the same node would "collide", and the node would not be able to grasp any of them. If, on the other hand, the messages represent visual or chemical signals (see e.g., [10,11,27,42]), then it may be unrealistic to assume that nodes cannot receive more than one of such messages at the same round and besides, by a standard balls-into-bins argument (e.g., by applying Lemma 3), it follows that in the uniform push model at each round no node receives more than O(log n) messages w.h.p. In this work we thus consider the model in which all messages are received, also because such assumption allows us to obtain simpler proofs than the other variants.…”

Section: The Reception Of Simultaneous Messagesmentioning

confidence: 99%

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Noisy rumor spreading and plurality consensus

Fraigniaud

Natale

2018

Distrib. Comput.

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Error-correcting codes are efficient methods for handling noisy communication channels in the context of technological networks. However, such elaborate methods differ a lot from the unsophisticated way biological entities are supposed to communicate. Yet, it has been recently shown by Feinerman et al. (PODC 2014) that complex coordination tasks such as rumor spreading and majority consensus can plausibly be achieved in biological systems subject to noisy communication channels, where every message transferred through a channel remains intact with small probability 1 2 + , without using coding techniques. This result is a considerable step towards a better understanding of the way biological entities may cooperate. It has nevertheless been established only in the case of 2-valued opinions: rumor spreading aims at broadcasting a single-bit opinion to all nodes, and majority consensus aims at leading all nodes to adopt the single-bit opinion that was initially present in the system with (relative) majority. In this paper, we extend this previous work to k-valued opinions, for any constant k ≥ 2. Our extension requires to address a series of important issues, some conceptual, others technical. We had to entirely revisit the notion of noise, for handling channels carrying k-valued messages. In fact, we precisely characterize the type of noise patterns for which plurality consensus is solvable. Also, a key result employed in the bivalued case by Feinerman et al. is an estimate of the probability of observing the most frequent opinion from observing the mode of a small sample. We generalize this result to the multivalued case by providing a new analytical proof for the bivalued case that is amenable to be extended, by induction, and that is of independent interest.

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Section: Other Related Worksupporting

confidence: 72%

Section: Other Related Workmentioning

confidence: 81%

Section: The Reception Of Simultaneous Messagesmentioning

confidence: 99%

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Noisy rumor spreading and plurality consensus

Fraigniaud

Natale

2018

Distrib. Comput.

Self Cite

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“…The so-called 3-majority dynamics for plurality consensus with k opinions was analyzed in [10]. In this protocol, each node samples three neighbors and adopts the majority opinion among the sample, breaking ties uniformly at random.…”

Section: Related Workmentioning

confidence: 99%

“…For a simple lower bound on synchronous protocols, we consider the classical synchronous model [9,10], where we assume that each node may communicate with O(polylog n) nodes per round. Additionally, we assume that the nodes do not know the set of initial opinions (however k may be known to the nodes).…”

Section: Breaking the Lower Bound On Synchronized Protocolsmentioning

confidence: 99%

Positive Aging Admits Fast Asynchronous Plurality Consensus

Bankhamer

Elsässer

Kaaser

et al. 2020

Proceedings of the 39th Symposium on Principles of Distributed Computing

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We study distributed plurality consensus among n nodes, each of which initially holds one of k opinions. The goal is to eventually agree on the initially dominant opinion. We consider an asynchronous communication model in which each node is equipped with a random clock. Whenever the clock of a node ticks, it may open communication channels to a constant number of other nodes, chosen uniformly at random or from a list of constantly many addresses acquired in previous steps. The tick rates and the delays for establishing communication channels (channel delays) follow some probability distribution. Once a channel is established, communication between nodes can be performed instantaneously. We consider distributions for the waiting times between ticks and channel delays that have constant mean and the so-called positive aging property. In this setting, asynchronous plurality consensus is fast: if the initial bias between the largest and second largest opinion is at least √ n log n, then after O (log log α k • log k + log log n) time all but a 1/polylog n fraction of nodes have the initial plurality opinion. Here α denotes the initial ratio between the largest and second largest opinion. After additional O (log n) steps all nodes have the same opinion w.h.p., and this result is tight. If additionally the distributions satisfy a certain density property, which is common in many well-known distributions, we show that consensus is reached in O (log log α k + log log n) time for all but n/polylog n nodes, w.h.p. This implies that for a large range of initial configurations partial consensus can be reached significantly faster in this asynchronous communication model than in the synchronous setting. To obtain these results, we first assume the existence of a designated base station and later present fully distributed algorithms. Additionally, we derive tail bounds on the Pólya-Eggenberger distribution, which might be of independent interest.

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Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

Shiraga

2021

Random Struct Algorithms

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This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the Best-of-two and the Best-of-three. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model G(2n, p, q), which is a random graph consisting of two distinct Erdős-Rényi graphs G(n, p) joined by random edges with density q ≤ p. We obtain two main results. First, if p = ω(log n/n) and r = q/p is a constant, we show that there is a phase transition in r with threshold r * (specifically, r * = √ 5 − 2 for the Best-of-two, and r * = 1/7 for the Best-of-three). If r > r * , the process reaches consensus within O(log log n + log n/ log(np)) steps for any initial opinion configuration with a bias of Ω(n). By contrast, if r < r * , then there exists an initial opinion configuration with a bias of Ω(n) from which the process requires at least 2 Ω(n) steps to reach consensus. Second, if p is a constant and r > r * , we show that, for any initial opinion configuration, the process reaches consensus within O(log n) steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

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Simple dynamics for plurality consensus

Cited by 44 publications

References 24 publications

Noisy rumor spreading and plurality consensus

Noisy rumor spreading and plurality consensus

Positive Aging Admits Fast Asynchronous Plurality Consensus

Phase transitions of Best‐of‐two and Best‐of‐three on stochastic block models

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