Probability laws of consensus in a broadcast-based consensus-forming algorithm

IF 0.5 4区数学 Q4 STATISTICS & PROBABILITY Stochastic Models Pub Date : 2021-10-13 DOI:10.1080/15326349.2021.1982394

Dai Katoh, S. Shioda

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引用次数: 2

Abstract

Abstract The consensus attained in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the probability laws of the consensus in a broadcast-based consensus-forming algorithm. First, we derive a fundamental equation on the time evolution of the opinions of agents. From the derived equation, we show that the consensus attained by the algorithm is given as a fixed-point solution of a linear equation. We then focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus with an algorithm for computing the distribution function of the consensus numerically. In the infinite-number-of-agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a Lévy distribution.

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基于广播的共识形成算法中共识的概率规律

摘要在共识形成算法中获得的共识通常不是常数，而是随机变量，即使初始意见相同。在本文中，我们研究了基于广播的一致性形成算法中一致性的概率规律。首先，我们推导了代理人意见的时间演化的基本方程。从导出的方程中，我们证明了该算法所获得的一致性是线性方程的不动点解。然后，我们关注两个极端情况：两个主体形成共识和无限多个主体形成一致。在双智能体的情况下，我们导出了一致性分布函数的几个性质，并用一种算法对一致性的分布函数进行了数值计算。在无穷多代理的情况下，我们证明了如果初始意见遵循稳定分布，那么共识也遵循稳定分布。此外，当初始意见遵循高斯分布、柯西分布或莱维分布时，我们导出了一致性的概率密度函数的闭合形式表达式。

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来源期刊

Stochastic Models 数学-统计学与概率论

CiteScore

1.30

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Stochastic Models publishes papers discussing the theory and applications of probability as they arise in the modeling of phenomena in the natural sciences, social sciences and technology. It presents novel contributions to mathematical theory, using structural, analytical, algorithmic or experimental approaches. In an interdisciplinary context, it discusses practical applications of stochastic models to diverse areas such as biology, computer science, telecommunications modeling, inventories and dams, reliability, storage, queueing theory, mathematical finance and operations research.