基于无约束未定态动力学的快速一致性
Fast Consensus via the Unconstrained Undecided State Dynamics
摘要：
我们考虑了$n$代理之间的多数共识问题。最初，每个代理具有$k$不同意见之一。智能体选择随机的交互伙伴，并根据自己的状态和交互伙伴的状态，根据一个固定的转换函数修改自己的状态。目标是达成一个共识配置，其中所有代理人都同意相同的意见，如果最初对一个意见有足够大的偏见，那么该意见应该占上风。我们分析了一个同步变量的未决定的状态动态定义如下。代理分阶段行动，包括决策和推进部分。在决策部分，任何遇到不同意见的代理人的代理人都变得犹豫不决。在助推部分，未决定的代理人采用他们遇到的第一个意见。我们在人口模型和流言模型中考虑这种动力学。对于群体模型，我们的协议达成共识（w. h. p.）在$O(\log^2 n)$并行时间中，提供$k > 2$（w. h. p.）in this model模型.在没有任何关于偏差的假设的情况下，仅针对未决定状态动态的非同步版本的$k = 2$示出了快速一致性[Clementi等人，MFCS'18]。我们表明，未决定的状态动力学的同步变体达到共识（w. h. p.）在时间$O(\log^2 n)$，独立于最初的数量，偏见，或意见的分布。在这两个模型中，我们保证，如果存在$Ω(\sqrt{n \log n})$的初始偏差，则（w. h. p.）这个观点赢了。我们的协议在八卦模型中的一个简单的扩展产生了一个动态，不依赖于$n$或$k$，是匿名的，并具有（w. h. p.）运行时$O(\log^2 n)$。这解决了Becchetti等人提出的一个开放问题。[分布式计算，~2017]。
英文摘要：
We consider the plurality consensus problem among $n$ agents. Initially, each agent has one of $k$ different opinions. Agents choose random interaction partners and revise their state according to a fixed transition function, depending on their own state and the state of the interaction partners. The goal is to reach a consensus configuration in which all agents agree on the same opinion, and if there is initially a sufficiently large bias towards one opinion, that opinion should prevail. We analyze a synchronized variant of the undecided state dynamics defined as follows. The agents act in phases, consisting of a decision and a boosting part. In the decision part, any agent that encounters an agent with a different opinion becomes undecided. In the boosting part, undecided agents adopt the first opinion they encounter. We consider this dynamics in the population model and the gossip model. For the population model, our protocol reaches consensus (w.h.p.) in $O(\log^2 n)$ parallel time, providing the first polylogarithmic result for $k > 2$ (w.h.p.) in this model. Without any assumption on the bias, fast consensus has only been shown for $k = 2$ for the unsynchronized version of the undecided state dynamics [Clementi et al., MFCS'18]. We show that the synchronized variant of the undecided state dynamics reaches consensus (w.h.p.) in time $O(\log^2 n)$, independently of the initial number, bias, or distribution of opinions. In both models, we guarantee that if there is an initial bias of $Ω(\sqrt{n \log n})$, then (w.h.p.) that opinion wins. A simple extension of our protocol in the gossip model yields a dynamics that does not depend on $n$ or $k$, is anonymous, and has (w.h.p.) runtime $O(\log^2 n)$. This solves an open problem formulated by Becchetti et al.~[Distributed Computing,~2017].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
