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A rigorous formulation of and partial results on Lorenz's "consensus strikes back" phenomenon for the Hegselmann-Krause model
arXiv - CS - Multiagent Systems Pub Date : 2021-07-26 , DOI: arxiv-2107.12906
Edvin Wedin

In a 2006 paper, Jan Lorenz observed a curious behaviour in numerical simulations of the Hegselmann-Krause model: Under some circumstances, making agents more closed-minded can produce a consensus from a dense configuration of opinions which otherwise leads to fragmentation. Suppose one considers initial opinions equally spaced on an interval of length $L$. As first observed by Lorenz, simulations suggest that there are three intervals $[0, L_1)$, $(L_1, L_2)$ and $(L_2, L_3)$, with $L_1 \approx 5.23$, $L_2 \approx 5.67$ and $L_3 \approx 6.84$ such that, when the number of agents is sufficiently large, consensus occurs in the first and third intervals, whereas for the second interval the system fragments into three clusters. In this paper, we prove consensus for $L \leq 5.2$ and for $L$ sufficiently close to 6. These proofs include large computations and in principle the set of $L$ for which consensus can be proven using our approach may be extended with the use of more computing power. We also prove that the set of $L$ for which consensus occurs is open. Moreover, we prove that, when consensus is assured for the equally spaced systems, this in turn implies asymptotic almost sure consensus for the same values of $L$ when initial opinions are drawn independently and uniformly at random. We thus conjecture a pair of phase transitions, making precise the formulation of Lorenz's "consensus strikes back" hypothesis. Our approach makes use of the continuous agent model introduced by Blondel, Hendrickx and Tsitsiklis. Indeed, one contribution of the paper is to provide a presentation of the relationships between the three different models with equally spaced, uniformly random and continuous agents, respectively, which is more rigorous than what can be found in the existing literature.

中文翻译:

对 Hegselmann-Krause 模型的 Lorenz“共识反击”现象的严格表述和部分结果

在 2006 年的一篇论文中,Jan Lorenz 在 Hegselmann-Krause 模型的数值模拟中观察到了一个奇怪的行为:在某些情况下,让代理人更加封闭可以从密集的意见配置中产生共识,否则会导致分裂。假设考虑在长度为 $L$ 的区间上等距分布的初始意见。正如 Lorenz 首先观察到的,模拟表明存在三个区间 $[0, L_1)$, $(L_1, L_2)$ 和 $(L_2, L_3)$,其中 $L_1 \approx 5.23$, $L_2 \approx 5.67 $ 和 $L_3 \approx 6.84$ 这样,当代理的数量足够大时,在第一个和第三个间隔中会发生共识,而在第二个间隔中,系统会分成三个集群。在本文中,我们证明了 $L \leq 5.2$ 和 $L$ 足够接近 6 的共识。这些证明包括大量计算,原则上可以使用我们的方法证明共识的 $L$ 集可以通过使用更多计算能力进行扩展。我们还证明了达成共识的 $L$ 集是开放的。此外,我们证明,当对等距系统确保共识时,这反过来意味着当初始意见是独立且随机地抽取时,对于相同的 $L$ 值,渐近几乎肯定的共识。因此,我们推测了一对相变,从而精确地表述了洛伦兹的“共识反击”假设。我们的方法利用了 Blondel、Hendrickx 和 Tsitsiklis 引入的连续代理模型。的确,
更新日期:2021-07-28
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