Continuous Opinion Dynamics under Bounded Confidence: A Survey
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paper_title: A Discrete Nonlinear and Non-Autonomous Model of Consensus Formation

paper_content:

Consensus formation among n experts is modeled as a positive discrete dynamical system in n dimensions. The well–known linear but non–autonomous model is extended to a nonlinear one admitting also various kinds of averaging beside the weighted arithmetic mean. For this model a sufficient condition for reaching a consensus is presented. As a special case consensus formation under bounded confidence is analyzed.
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paper_title: Modelling collective opinion formation by means of active Brownian particles

paper_content:

The concept of active Brownian particles is used to model a collective opinion formation process. It is assumed that individuals in community create a two-component communication field that influences the change of opinions of other persons and/or can induce their migration. The communication field is described by a reaction-diffusion equation, the opinion change of the individuals is given by a master equation, while the migration is described by a set of Langevin equations, coupled by the communication field. In the mean-field limit holding for fast communication we derive a critical population size, above which the community separates into a majority and a minority with opposite opinions. The existence of external support (e.g. from mass media) changes the ratio between minority and majority, until above a critical external support the supported subpopulation exists always as a majority. Spatial effects lead to two critical "social" temperatures, between which the community exists in a metastable state, thus fluctuations below a certain critical wave number may result in a spatial opinion separation. The range of metastability is particularly determined by a parameter characterizing the individual response to the communication field. In our discussion, we draw analogies to phase transitions in physical systems. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.65.+b Self- organized systems - 87.23.Ge Dynamics of social systems
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paper_title: A stabilization theorem for dynamics of continuous opinions

paper_content:

A stabilization theorem for processes of opinion dynamics is presented. The theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging (like the models of Hegselmann-Krause and Weisbuch-Deffuant). The analysis detects self-confidence as a driving force of stabilization.
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paper_title: Communication regimes in opinion dynamics: Changing the number of communicating agents

paper_content:

This article contributes in four ways to the research on time-discrete continuous opinion dynamics with compromising agents. First, communication regimes are introduced as an elementary concept of opinion dynamic models. Second, we develop a model that covers two major models of continuous opinion dynamics, i.e. the basic model of Deffuant and Weisbuch as well as the model of Krause and Hegselmann. To combine these models, which handle different numbers of communicating agents, we convert the convergence parameter of Deffuant and Weisbuch into a parameter called self-support. Third, we present simulation results that shed light on how the number of communicating agents but also how the self-support affect opinion dynamics. The fourth contribution is a theoretically driven criterion when to stop a simulation and how to extrapolate to infinite many steps.
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paper_title: How can extremism prevail? A study based on the relative agreement interaction model

paper_content:

Abstract: We model opinion dynamics in populations of agents with continuous opinion and uncertainty. The opinions and uncertainties are modified by random pair interactions. We propose a new model of interactions, called relative agreement model, which is a variant of the previously discussed bounded confidence. In this model, uncertainty as well as opinion can be modified by interactions. We introduce extremist agents by attributing a much lower uncertainty (and thus higher persuasion) to a small proportion of agents at the extremes of the opinion distribution. We study the evolution of the opinion distribution submitted to the relative agreement model. Depending upon the choice of parameters, the extremists can have a very local influence or attract the whole population. We propose a qualitative analysis of the convergence process based on a local field notion. The genericity of the observed results is tested on several variants of the bounded confidence model.
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paper_title: Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

paper_content:

The agent-based bounded confidence model of opinion dynamics of Hegselmann and Krause (2002) is reformulated as an interactive Markov chain. This abstracts from individual agents to a population model which gives a good view on the underlying attractive states of continuous opinion dynamics. We mutually analyse the agent-based model and the interactive Markov chain with a focus on the number of agents and onesided dynamics. Finally, we compute animated bifurcation diagrams that give an overview about the dynamical behavior. They show an interesting phenomenon when we lower the bound of confidence: After the first bifurcation from consensus to polarisation consensus strikes back for a while.
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paper_title: Continuous Opinion Dynamics: Insights through Interactive Markov Chains

paper_content:

We reformulate the agent-based opinion dynamics models of Weisbuch-Deffuant and Hegselmann-Krause as interactive Markov chains. So we switch the scope from a finite number of n agents to a finite number of n opinion classes. Thus, we will look at an infinite population distributed to opinion classes instead of agents with real number opinions. The interactive Markov chains show similar dynamical behavior as the agent-based models: stabilization and clustering. Our framework leads to a discrete bifurcation diagram for each model which gives a good view on the driving forces and the attractive states of the system. The analysis shows that the emergence of minor clusters in the Weisbuch-Deffuant model and of meta-stable states with very slow convergence to consensus in the Hegselmann Krause model are intrinsic to the dynamical behavior.
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paper_title: Better Being Third Than Second In A Search For A Majority Opinion

paper_content:

Monte Carlo simulations of a Sznajd model show that if a near-consensus is formed out of four initially equally widespread opinions, the one which at intermediate times is second in the number of adherents usually loses out against the third-placed opinion.
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paper_title: Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

paper_content:

The agent-based bounded confidence model of opinion dynamics of Hegselmann and Krause (2002) is reformulated as an interactive Markov chain. This abstracts from individual agents to a population model which gives a good view on the underlying attractive states of continuous opinion dynamics. We mutually analyse the agent-based model and the interactive Markov chain with a focus on the number of agents and onesided dynamics. Finally, we compute animated bifurcation diagrams that give an overview about the dynamical behavior. They show an interesting phenomenon when we lower the bound of confidence: After the first bifurcation from consensus to polarisation consensus strikes back for a while.
---
paper_title: Continuous Opinion Dynamics: Insights through Interactive Markov Chains

paper_content:

We reformulate the agent-based opinion dynamics models of Weisbuch-Deffuant and Hegselmann-Krause as interactive Markov chains. So we switch the scope from a finite number of n agents to a finite number of n opinion classes. Thus, we will look at an infinite population distributed to opinion classes instead of agents with real number opinions. The interactive Markov chains show similar dynamical behavior as the agent-based models: stabilization and clustering. Our framework leads to a discrete bifurcation diagram for each model which gives a good view on the driving forces and the attractive states of the system. The analysis shows that the emergence of minor clusters in the Weisbuch-Deffuant model and of meta-stable states with very slow convergence to consensus in the Hegselmann Krause model are intrinsic to the dynamical behavior.
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paper_title: Dynamics of structured attitudes and opinions

paper_content:

Opinion dynamic models frequently assume an opinion to be a onedimensional concept and only sometimes vectors of opinions are considered. Contrary, psychological attitude theory considers an attitude to be a multidimensional concept, where each dimension is composed of beliefs and evaluative components. In this article we extend classic basic models of opinion dynamics such that they can capture this multidimensionality. We present models where individuals talk about difierent issues, e.g. evaluations or beliefs about the relation between objects and attributes. We present reinterpretations the results of previous simulations and we present results of simulations of new dynamics. Those new dynamics are based on a hierarchical approach to bounded confldence.
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paper_title: Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

paper_content:

The agent-based bounded confidence model of opinion dynamics of Hegselmann and Krause (2002) is reformulated as an interactive Markov chain. This abstracts from individual agents to a population model which gives a good view on the underlying attractive states of continuous opinion dynamics. We mutually analyse the agent-based model and the interactive Markov chain with a focus on the number of agents and onesided dynamics. Finally, we compute animated bifurcation diagrams that give an overview about the dynamical behavior. They show an interesting phenomenon when we lower the bound of confidence: After the first bifurcation from consensus to polarisation consensus strikes back for a while.
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paper_title: Compromise and Synchronization in Opinion Dynamics

paper_content:

Abstract.We discuss two models of opinion dynamics. We first present a brief review of ::: the Hegselmann and Krause (HK) compromise model in two dimensions, ::: showing that it is possible to simulate the dynamics ::: in the limit of an infinite number of agents by solving numerically a rate equation for ::: a continuum distribution of opinions. Then, we discuss the Opinion Changing Rate (OCR) model, ::: which allows to study under which conditions a group of agents with a ::: different natural tendency (rate) to change opinion can find the ::: agreement. In the context of the this model, consensus is viewed as a synchronization ::: process.
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paper_title: Modelling Group Opinion Shift to Extreme : the Smooth Bounded Confidence Model

paper_content:

We consider the phenomenon of opinion shift to the extreme reported in the social psychology literature. We argue that a good candidate to model this phenomenon can be a new variant of the bounded confidence (BC) model, the smooth BC model which we propose in this paper. This model considers individuals with a continuous opinion and an uncertainty. Individuals interact by random pairs, and attract each other's opinion proportionally to a Gaussian function of the distance between their opinions. We first show that this model presents a shift to the extreme when we introduce extremists (very convinced individuals with extreme opinions) in the population, even if there is the same number of extremists located at each extreme. This behaviour is similar to the one already identified with other versions of BC model. Then we propose a modification of the smooth BC model to account for the social psychology data and theories related to this phenomenon. The modification is based on the hypothesis of perspective taking (empathy) in the context of consensus seeking.
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paper_title: MONTE CARLO SIMULATION OF DEFFUANT OPINION DYNAMICS WITH QUALITY DIFFERENCES

paper_content:

In this work, the consequences of different opinion qualities in the Deffuant model were examined. If these qualities are randomly distributed, no different behavior was observed. In contrast to that, systematically assigned qualities had strong effects to the final opinion distribution. There was a high probability that the strongest opinion was one with a high quality. Furthermore, under the same conditions, this major opinion was much stronger than in the models without systematic differences. Finally, a society with systematic quality differences needed more tolerance to form a complete consensus than one without or with unsystematic ones.
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paper_title: Truth and Cognitive Division of Labour First Steps towards a Computer Aided Social Epistemology

paper_content:

The paper analyzes the chances for the truth to be found and broadly accepted under conditions of cognitive division of labour combined with a social exchange process. Cognitive division of labour means, that only some individuals are active truth seekers, possibly with different capacities. The social exchange process consists in an exchange of opinions between all individuals, whether truth seekers or not. We de- velop a model which is investigated by both, mathematical tools and computer simulations. As an analytical result the Funnel theorem states that under rather weak conditions on the social process a consensus on the truth will be reached if all individuals posses an arbitrarily small inclination for truth seeking. The Leading the pack theorem states that under certain conditions even a single truth seeker may lead all individuals to the truth. Systematic simulations analyze how close and how fast groups can get to the truth depending on the frequency of truth seekers, their capacities as truth seekers, the position of the truth (more to the extreme or more in the centre of an opinion space), and the willingness to take into account the opinions of others when exchanging and updating opinions. A tricky movie visualizes simulations results in a parameter space of higher dimensions.
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paper_title: Minorities in a Model for Opinion Formation

paper_content:

We study a model for social influence in which the agents' opinion is a continuous variable [G. Weisbuch et al., Complexity \textbf{7}, 2, 55 (2002)]. The convergent opinion adjustment process takes place as a result of random binary encounters whenever the difference between agents' opinions is below a given threshold. The inhomogeneity in the dynamics gives rise to a complex steady state structure, which is also highly dependent on the threshold and on the convergence parameter of the model.
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paper_title: A pr 2 00 4 The role of network topology on extremism propagation with the Relative Agreement opinion dynamics

paper_content:

In Deffuant et al. (J. Artif. Soc. Soc. Simulation 5 (2002) 4), we proposed a simple model of opinion dynamics, which we used to simulate the influence of extremists in a population. Simulations were run without any specific interaction structure and varying the simulation parameters, we observed different attractors such as predominance of centrism or of extremism. We even observed in certain conditions, that the whole population drifts to one extreme of the opinion, even if initially there are an equal number of extremists at each extreme of the opinion axis. In the present paper, we study the influence of the social networks on the presence of such a dynamical behavior. In particular, we use small-world networks with variable connectivity and randomness of the connections. We find that the drift to a single extreme appears only beyond a critical level of connectivity, which decreases when the randomness increases.
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paper_title: Communication regimes in opinion dynamics: Changing the number of communicating agents

paper_content:

This article contributes in four ways to the research on time-discrete continuous opinion dynamics with compromising agents. First, communication regimes are introduced as an elementary concept of opinion dynamic models. Second, we develop a model that covers two major models of continuous opinion dynamics, i.e. the basic model of Deffuant and Weisbuch as well as the model of Krause and Hegselmann. To combine these models, which handle different numbers of communicating agents, we convert the convergence parameter of Deffuant and Weisbuch into a parameter called self-support. Third, we present simulation results that shed light on how the number of communicating agents but also how the self-support affect opinion dynamics. The fourth contribution is a theoretically driven criterion when to stop a simulation and how to extrapolate to infinite many steps.
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paper_title: Attitude Dynamics With Limited Verbalisation Capabilities

paper_content:

This article offers a new perspective for research on opinion dynamics. It demonstrates the importance of the distinction of opinion and attitude, which originally has been discussed in literature on consumer behaviour. As opinions are verbalised attitudes not only biases in interpretation and adoption processes have to be considered but also verbalisation biases should be addressed. Such biases can be caused by language deficits or social norms. The model presented in this article captures the basic features of common opinion dynamic models and additionally biases in the verbalisation process. Further, it gives a first analysis of this model and shows that precision as bias in the verbalisation process can influence the dynamics significantly. Presenting and applying the concept of area of influential attitudes the impact of each parameter (selective attitude, selective interpretation, and precision) is analysed independently. Some preliminary results for combined effects are presented.
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paper_title: How can extremism prevail? A study based on the relative agreement interaction model

paper_content:

Abstract: We model opinion dynamics in populations of agents with continuous opinion and uncertainty. The opinions and uncertainties are modified by random pair interactions. We propose a new model of interactions, called relative agreement model, which is a variant of the previously discussed bounded confidence. In this model, uncertainty as well as opinion can be modified by interactions. We introduce extremist agents by attributing a much lower uncertainty (and thus higher persuasion) to a small proportion of agents at the extremes of the opinion distribution. We study the evolution of the opinion distribution submitted to the relative agreement model. Depending upon the choice of parameters, the extremists can have a very local influence or attract the whole population. We propose a qualitative analysis of the convergence process based on a local field notion. The genericity of the observed results is tested on several variants of the bounded confidence model.
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paper_title: Opinion Dynamics Driven by Various Ways of Averaging

paper_content:

The paper treats opinion dynamics under bounded confidence when agents employ, beside an arithmetic mean, means like a geometric mean, a power mean or a random mean in aggregating opinions. The different kinds of collective dynamics resulting from these various ways of averaging are studied and compared by simulations. Particular attention is given to the random mean which is a new concept introduced in this paper. All those concrete means are just particular cases of a partial abstract mean, which also is a new concept. This comprehensive concept of averaging opinions is investigated also analytically and it is shown in particular, that the dynamics driven by it always stabilizes in a certain pattern of opinions.
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Title: Continuous Opinion Dynamics under Bounded Confidence: A Survey

Section 1: Introduction
Description 1: Write an introductory overview discussing the emergence of opinion dynamics, the nature of continuous opinion dynamics, and relevant models and their historical contexts.

Section 2: The models
Description 2: Describe the two main models of continuous opinion dynamics (Deffuant-Weisbuch and Hegselmann-Krause) in both their agent-based and density-based formulations, including definitions and key processes.

Section 3: The Deffuant-Weisbuch model
Description 3: Provide a detailed explanation of the Deffuant-Weisbuch (DW) model, including its agent-based and density-based versions, mathematical formulations, and behavior under different conditions.

Section 4: The Hegselmann-Krause model
Description 4: Offer a thorough description of the Hegselmann-Krause (HK) model, outlining its agent-based and density-based versions, mathematical formulations, and unique characteristics.

Section 5: Bifurcation Diagrams
Description 5: Present and explain the bifurcation diagrams for both homogeneous density-based models, highlighting the location and transitions of clusters concerning the bound of confidence.

Section 6: Extensions
Description 6: Review various extensions to the basic continuous opinion dynamics models, including multidimensional opinions, heterogeneous bounds of confidence, social networks, different communication regimes, and other miscellaneous factors.

Section 7: Conclusions and open problems
Description 7: Summarize the key findings, highlight the importance of these models, and discuss the remaining open problems and future research directions in the field of continuous opinion dynamics.