# Tight Analysis for the 3-Majority Consensus Dynamics

@article{Ghaffari2017TightAF, title={Tight Analysis for the 3-Majority Consensus Dynamics}, author={Mohsen Ghaffari and Johannes Lengler}, journal={ArXiv}, year={2017}, volume={abs/1705.05583} }

We present a tight analysis for the well-studied randomized 3-majority dynamics of stabilizing consensus, hence answering the main open question of Becchetti et al. [SODA'16].
Consider a distributed system of n nodes, each initially holding an opinion in {1, 2, ..., k}. The system should converge to a setting where all (non-corrupted) nodes hold the same opinion. This consensus opinion should be \emph{valid}, meaning that it should be among the initially supported opinions, and the (fast…

## 9 Citations

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It is proved that, starting from any initial configuration, the process reaches a monochromatic configuration within O(log n) rounds, with high probability, and this bound turns out to be tight.

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This analysis shows that, starting from any initial configuration, the Undecided-State Dynamics reaches a monochromatic configuration within $O(\log^2 n)$ rounds, with high probability, and proves that if the initial configuration has bias $\Omega(\sqrt{n\log n})$, then the dynamics converges toward the initial majority color within a polylogarithmic number of rounds.

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It is shown that the following performance can be achieved on a $d-regular expander using two-sample voting on general expanders in the case of three or more opinions: the largest opinion wins in O((n \log n)/A_1) steps with high probability.

### Phase transition of the 2-Choices dynamics on core–periphery networks

- Computer ScienceDistributed Computing
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By looking at the 2-Choices dynamics as a simplistic model of competition among opinions in social networks, a theorem sheds light on the influence of the core on the rest of the network, as a function of thecore's connectivity towards the latter.

### Probabilistic analysis of distributed processes with focus on consensus. (Analyse probabiliste de processus distribués axés sur les processus de consensus)

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This thesis takes the rich tool set from probability theory for the analysis of Markov Chains and employs it to study a wide range of distributed processes: Forest Fire Model, Balls-into-Bins with Deleting Bins, and fundamental consensus dynamics and protocols such as the Voter Model, 2-Choices, and 3-Majority.

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It is proved that for any graph the coalescence time is bounded by O(n^3) (which is tight for the Barbell graph); surprisingly even such a basic question about the coalescing time was not answered before this work.

### On the Necessary Memory to Compute the Plurality in Multi-Agent Systems

- MathematicsCIAC
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This work considers the Relative-Majority Problem (also known as Plurality) in the general Population Protocols model in which, given an underlying undirected connected graph whose nodes represent the agents, edges are selected by a globally fair scheduler.

### 43rd International Symposium on Mathematical Foundations of Computer Science: MFCS 2018, August 27-31, 2018, Liverpool, United Kingdom

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