Nearly-Tight Analysis for 2-Choice and 3-Majority Consensus Dynamics

Ghaffari, Mohsen; Lengler, Johannes

doi:10.1145/3212734.3212738

Cited by 35 publications

(53 citation statements)

References 18 publications

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“…They considered q initial different opinions and proved that if the initial imbalance in the number of opinions between the first and second majorities is Ω(min{ √ 2q, (n/ log n) 1/6 } n/ log n), then consensus is reached with high probability in O(min{q, (n/ log n) 1/3 } log n) steps on the initial majority. Similar results can be proved for Best-of-2 dynamics [8].…”

Section: Introductionsupporting

confidence: 84%

Best-of-Three Voting on Dense Graphs

Kang

Rivera

2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Bestof-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n α , where α = Ω (log log n) −1 . We prove that if initially each vertex is red with probability greater than 1/2 + δ, and blue otherwise, where δ ≥ (log d) −C for some C > 0, then with high probability this dynamic reaches a final state where all vertices are red within O (log log n) + O log δ −1 steps .

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Section: Introductionsupporting

confidence: 84%

Best-of-Three Voting on Dense Graphs

Kang

Rivera

2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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“…In this setting, opinion dynamics are mathematical models to investigate the way a fully-decentralized MAS is able to reach some form of Consensus. Their study is a hot topic touching several research areas such as MAS [15,21], Distributed Computing [7,20,29], Social Networks [1,38], and System Biology [11,12]. Typical examples of opinion dynamics are the Voter Model, the averaging rules, and the majority rules.…”

Section: Introductionmentioning

confidence: 99%

“…Typical examples of opinion dynamics are the Voter Model, the averaging rules, and the majority rules. Some of such dynamics share a surprising efficiency and resiliency that seem to exploit common computational principles [7,20,29].…”

Section: Introductionmentioning

confidence: 99%

Phase Transition of a Non-linear Opinion Dynamics with Noisy Interactions

d'Amore

Clementi

Natale

2020

Structural Information and Communication Complexity

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In several real Multi-Agent Systems (MAS), it has been observed that only weaker forms of metastable consensus are achieved, in which a large majority of agents agree on some opinion while other opinions continue to be supported by a (small) minority of agents. In this work, we take a step towards the investigation of metastable consensus for complex (non-linear) opinion dynamics by considering the famous Undecided-State dynamics in the binary setting, which is known to reach consensus exponentially faster than the Voter dynamics. We propose a simple form of uniform noise in which each message can change to another one with probability p and we prove that the persistence of a metastable consensus undergoes a phase transition for p = 1 6. In detail, below this threshold, we prove the system reaches with high probability a metastable regime where a large majority of agents keeps supporting the same opinion for polynomial time. Moreover, this opinion turns out to be the initial majority opinion, whenever the initial bias is slightly larger than its standard deviation. On the contrary, above the threshold, we show that the information about the initial majority opinion is "lost" within logarithmic time even when the initial bias is maximum. Interestingly, using a simple coupling argument, we show the equivalence between our noisy model above and the model where a subset of agents behave in a stubborn way.

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“…More recently, a detailed study and comparison of the 3-majority dynamics and the related two-choices process has been performed by [12]. Subsequently, a tight analysis of these processes was presented in [27]. Together, [27] and [12] cover a large range of parameters k.…”

Section: Related Workmentioning

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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Nearly-Tight Analysis for 2-Choice and 3-Majority Consensus Dynamics

Cited by 35 publications

References 18 publications

Best-of-Three Voting on Dense Graphs

Best-of-Three Voting on Dense Graphs

Phase Transition of a Non-linear Opinion Dynamics with Noisy Interactions

Positive Aging Admits Fast Asynchronous Plurality Consensus

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