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

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.

Keywords:

• Broadcast
• consensus
• stable distribution
• Gaussian
• Cauchy
• Lévy

Notes

1 The present paper is an extended version of the paper presented in IFIP WG 7.3 Performance 2020 [14].

2 Because of the irreducibility, there exists $n\in \mathbb{N}$ such that the elements of Qⁿ are all positive, even if Q has a non-positive element. Thus, all results (Lemma 3.1, Lemma 3.2, and Theorem 3.3) of Section 2 hold by using Qⁿ instead of Q, even if Q has non-positive elements. In Sections 4–6, we assume that all agents can directly communicate with each other and $w\in (0,1),$ so Q does not have a non-positive element.

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science (JSPS) through KAKENHI Grant Numbers JP19H02135, JP19H04096, and JP20K21783.