Consensus formation under bounded confidence

At the beginning of this century, Hegselmann and Krause proposed a dynamical model for opinion formation that is referred to as the Bounded Confidence Opinion Dynamics (BCOD) model and that has since attracted a wide interest from different research communities. The model can be viewed as a dynamic network, in which each agent is endowed with a state variable representing an opinion and two agents interact if the distance between their opinions does not exceed a constant confidence bound. This relation of instantaneous proximity between the opinions naturally induces a dynamic interaction graph. At each stage of the opinion iteration, all agents synchronously update their opinion to the average of all opinions that belong to the neighbors in the interaction graph.

BCOD models exhibit a broad variety of phenomena that cannot be studied by traditional methods, and their analysis has enriched the systems and control field with a number of novel mathematical tools. This fact, together with the existence of an extensive literature on the topic scattered across different fields, calls for a systematic presentation of the existing results on this class of dynamic networks. The aim of this survey is to provide an overview of BCOD models with time-synchronous interactions, with possibly asymmetric and heterogeneous confidence bounds. Conditions on the different classes of BCOD which ensure the convergence (in finite time or asymptotically) of the opinions are discussed, and the possible structures of the terminal opinions are described. The numerous phenomena highlighted in the literature from numerical studies, e.g., the characterization of steady state behaviors and the sensitivity to confidence thresholds, are also reviewed. Finally, some recent modifications and applications of BCOD models are discussed, and suggestions of directions for future research are provided.

Recently, belief system dynamics has been proposed to model the opinion evolution of a social group on a set of logically interdependent topics (captured by a logic matrix). To analyze the evolution of belief system after a social group embeds into another social group when the two social groups have different ideologies on the logic dependence of topics, we creatively propose an extended multidimensional DeGroot model, which is modeled as the co-evolution of belief systems of two factions under social embedding, where the logic matrices of the two factions are considered as coherently heterogeneous and non-coherently heterogeneous. Through strict theoretical analysis for the proposed model, we demonstrate that social embedding can change a faction’s belief system and quantify the effect under the two heterogeneities. Then, based on social embedding, we propose an embedding method of social robot to guide online public opinion. This method achieves the purpose of controlling online public opinion by establishing one-way communication from leaders in the social network to the social robot and adaptively controlling the belief system of the social robot. To verify the theoretical analysis results and supplement some new findings, we perform simulations to explore the impact of the number of social embeddings and the self-confidence degree of the social group on the time required for the belief system to reach stable state.

Consensus reaching is an iterative and dynamic process that supports group decision-making models by guiding decision-makers towards modifying their opinions through a feedback mechanism. Many attempts have been recently devoted to the design of efficient consensus reaching processes, especially when the dynamism is dependent on time, which aims to deal with opinion dynamics models. The emergence of novel methodologies in this field has been accelerated over recent years. In this regard, the present work is concerned with a systematic review of classical dynamic consensus and opinion dynamics models. The most recent trends of both models are identified and the developed methodologies are described in detail. Challenges of each model and open problems are discussed and worthwhile directions for future research are given. Our findings denote that due to technological advancements, a majority of recent literature works are concerned with the large-scale group decision-making models, where the interactions of decision-makers are enabled via social networks. Managing the behavior of decision-makers and consensus reaching with the minimum adjustment cost under social network analysis have been the top priorities for researchers in the design of classical consensus and opinion dynamics models.

Bounded confidence opinion dynamics are dynamic networks in which agents are connected if their opinions are similar, and each agent updates her opinion as the average of the neighbors’ opinions. In homogeneous asymmetric Heglselmann–Krause (HK) models, all agents have the same confidence thresholds which could be different for the selection of upper and lower neighbors. This paper provides conditions for the convergence of the opinions to consensus and to clustering for this class of HK models. A new tighter bound on the time interval for reaching the steady state is also provided.

Failure mode and effect analysis (FMEA) is a preventive risk management tool, and has been widely applied across various industries. Most existing FMEA methods are concentrated on obtaining a complete risk ranking of failure modes. However, in some situations, it may be enough to divide failure modes into several ordinal classifications. Moreover, the consensus issue needs to be taken seriously because multiple FMEA experts have different risk assessments due to their diverse knowledge backgrounds. Meanwhile, bounded confidences, opinion evolution and psychological behaviours of FMEA experts also play key roles in the consensus-reaching process. Therefore, this paper integrates a classification consensus-based model with bounded confidences and opinion evolution, from the perspective of prospect theory into FMEA. First, FMEA experts provide risk assessments by prospect theory. Then, a novel consensus-reaching process is developed, in which a dual feedback adjustment mechanism based on opinion evolution and bounded confidences is designed. Finally, a case study and a comparative analysis are provided to show the application value of the proposal. The classification consensus-based proposal enriches the theory system of FMEA, and has a good reference value for solving the reliability management problems in the aerospace, automobile manufacturing and other fields.

Self-confidence actually affects group decision making (GDM) because it can somehow reflect decision maker (DM)’s expertise and cognitive level. This paper proposes a new form of preference relation named interval-valued fuzzy preference relation with self-confidence (IFPR-SC). IFPR-SC requires DMs to express self-confidence levels when offering interval-valued preferences. Some operation laws regarding IFPR-SC are presented. Moreover, this paper uses an induced-ordered weighted aggregation operator to aggregate individual IFPRs-SC, then presents a novel feedback adjustment mechanism called improved Hegselmann-Kraus (IHK) model to update DMs’ IFPRs-SC. Finally, this paper offers an illustrative example and several comparisons to verify the feasibility and efficiency of IHK model.