小置信度下远程意见模型的共识概率研究

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本文主要探讨了"具有小置信度的远程意见模型的共识分析"这一研究主题。在这个模型中，作者 Jiangbo Zhang 和 Yiguang Hong 拓展了传统意见动力学的研究领域，关注的是在长期范围内的意见交互如何影响意见的一致性。每个个体（代理）通过考虑其他个体的意见值及其自身的置信区间来更新其观点。核心概念是置信边界，即个体在接收信息时所设定的信任程度阈值。作者提出了一种带有限置信度的意见模型，当置信边界足够小的时候，他们着重于计算在这种情况下达成意见一致的概率下限。这不仅有助于理解在高度不确定的信息环境中，群体决策如何趋于稳定，也为理解和控制复杂社会系统中的信息传播提供了理论基础。研究中，他们对意见随时间和可调整参数的变化进行了深入讨论，通过模拟示例展示了这些参数如何影响最终的共识概率。此外，文章还涉及到了几个关键术语，如"opinion dynamics"（意见动力学）、"consensus probability"（共识概率）以及"long-range interaction"（远程交互），这些都是理解和分析此类模型不可或缺的概念。这项工作对于理解社交网络中的信息扩散、群体决策行为以及社会影响力的研究具有重要意义，也为未来的多智能体系统和复杂网络理论提供了新的思考视角。通过这个研究，读者可以了解到在信息不对称和信任度有限的情况下，如何通过理论分析预测和优化社会系统中的意见一致性。

Consensus Analysis of A Long-Range Opinion Model with A

Small Conﬁdence Bound

Jiangbo Zhang

, Yiguang Hong

1. Jiangbo Zhang is with the faculty of science, Southwest Petroleum University; Chengdu 610500, P. R. China

2. Key Lab of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,

P. R. China,

E-mail: jbzhang@amss.ac.cn; yghong@iss.ac.cn

Abstract: In this paper, we propose a bounded-conﬁdence opinion model and focus on the study of the opinion consensus prob-

ability based on long-range opinion interactions. In this model, each agent updates its opinion using some agents’ opinion values

according to its conﬁdence bound. We provide a lower bound of the opinion consensus probability when the conﬁdence bound

is sufﬁciently small. Also, we give discussion and simulation examples for the opinion evolution along with some adjustable

parameters.

Key Words: opinion dynamics, consensus probability, conﬁdence bound, long-range interaction

1 Introduction

In recent years, social problems have attracted more and

more attention for both the academy and public, and the

modeling and analysis of social opinion dynamics become

an important theoretical research direction to understand the

underlying social mechanism (referring to [4, 6, 11, 12]).

In light of multi-agent networks, different models for such

problems have been proposed [2, 8, 10, 13, 21], and vari-

ous mathematical methods, such as graph theory, stochastic

process, optimization, and control theory, have been used to

analyze them [18, 22].

As we know, many existing opinion dynamics models il-

lustratively showed how the opinions reach their agreement.

[1, 24]. Based on the zero-entries symmetry of the adjacency

matrices, [5] proved the stability of one class of agent-based

opinion dynamics models, including the discrete-time ho-

mogeneous Hegselmann-Krause (HK) model and the sym-

metric homogeneous Deffuant-Weisbuch (DW) model [19].

Then [6] developed simple methods to show the convergence

of one kind of bound conﬁdence model, while [7] discussed

empirical opinion density using the system’s mean-ﬁeld dy-

namics as the population size of the agents becomes sufﬁ-

ciently large. From the viewpoint of inﬁnite ﬂow stability of

certain random matrices, [3] provided a new proof about the

convergence of the HK model and gave a convergence ter-

mination time bound for this model. Recently, [16] analyzed

the “Group-think” problem by studying the cooperation of

the appraise dynamics and inﬂuence dynamics. [14] stud-

ied some opinion models such as French-DeGroot model,

the linear Abelson model, Friedkin-Johnsen model and Tay-

lor model, with graph-based methods to show how opinion

states evolve in the networks.

Consensus and convergence of multi-agent systems have

been studied widely [9–11, 13, 15]. However, many tech-

niques for the analyze of multi-agent systems (with switch-

ing topologies described by a given random switching signal

This work was supported by the project of Systems science and enter-

prise development, Sichuan Province (Grant No. XQ15B10), the scientiﬁc

research starting project of SWPU (Grant No. 2014QHZ032), the NNSF

of China (Grant Nos. 61333001, 61573344), and Beijing Natural Science

Foundation (Grant No. 4152057).

function [12]) cannot apply to the bounded conﬁdence mod-

el, because the switching topologies are state-dependent.

In this paper, we study the consensus probability of one

bounded-conﬁdence opinion model- the Long-range mod-

el [23] with a small conﬁdence bound. The motivation of

the research is to character the consensus probability for a

proposed bounded-conﬁdence opinion model. The key fea-

tures of our model include its bounded conﬁdence restric-

tions (the acceptable degree of the regular agent opinions)

and the multi-selection long-range learning [17]. Although

our model can be viewed as a Markov process, the meth-

ods for graph-based network analysis and stochastic ma-

trix analysis cannot be applied to this long-rang interac-

tion model, because this conﬁdence-based model is state-

dependent and highly nonlinear resulting from the bounded

conﬁdence assigned to each regular agent. Other methods

used in the study of random multi-agent coordination (refer-

ring to [13, 22]) are not effective in our problem, either. The

main contributions in this paper include that we introduce an

bounded conﬁdence opinion model with multiple selection

methods, and provide the estimation for the lower bound for

the probability of opinion consensus in order to measure the

opinions’ aggregation degree, when the conﬁdence bound is

sufﬁciently small.

The rest of this paper is organized as follows. Section 2 in-

troduces preliminary background and formulates the opinion

model. Section 3 provides the lower bound of agent opinion

limit states’ consensus probability related with some param-

eters and demonstrates the consensus probability when the

conﬁdence bound is sufﬁciently small. The Section 4 shows

some numerical simulations. Finally, Section 5 gives some

concluding remarks.

Notations: N denotes the non-negative integer number set

{0, 1, 2,...}, while N

is the positive integer number set

{1, 2,...}; R is the real number set; R

m×n

means the m×n

matrix with entries taking values in R; 1

m×n

and 0

m×n

de-

note the m × n matrix with all entries being 1 and 0, respec-

tively.

Proceedings of the 36th Chinese Control Conference

Jul

26-28, 2017, Dalian, China

1533

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