Bounded Confidence Opinion Dynamics Research Articles

Effects of correlation between bounded confidences and node degrees on consensus in opinion dynamics on complex networks

In bounded confidence opinion dynamics, agents only consider other agents when their opinions are sufficiently close. In many works, the confidence bound is assumed as a homogeneous one. However, agents in different situations may be with different confidence bounds in real life. In this work, the correlation of degrees and confidence bounds is introduced into modified Hegselmann–Krause model on complex networks and we mainly focus on whether the population can reach consensus, where all agents hold a same opinion. The number of opinion clusters, the relative size of the largest cluster and the probability of reaching consensus are the measures of the opinion cluster profile. The numerical simulation shows that the existence of the correlation promotes the probability of reaching consensus and even in the case where some agents are with very small confidence bounds. On scale-free and Erdös–Rényi random networks, the negative correlation supports the consensus much more than the positive correlation case. However, two correlations show the same effect on small-world networks, and this effect is also observed in scale-free and Erdös–Rényi random networks when the average degree becomes very large.

Phase coexistence in the fully heterogeneous Hegselmann-Krause opinion dynamics model.

We present an extensive study of the joint effects of heterogeneous social agents and their heterogeneous social links in a bounded confidence opinion dynamics model. The full phase diagram of the model is explored for two different network's topologies and compared to two opposed extreme cases: on one hand, the heterogeneous agents constitute a mixed population and on the other, their interactions are modeled by a lattice. The results show that when agents prone to compromise coexist with close-minded ones, the steady state of the dynamics shows coexistent phases. In particular, unlike the case of homogeneous agents in networks, or heterogeneous agents in a fully mixed population, it is possible that the society ends up in consensus around one extreme opinion. Moreover, during the dynamics, the consensus may be overturned from one extreme to the other of the opinion space. We also show that the standard order parameter, the normalized average size of the largest opinion cluster, may be misleading in this case, as it hides the existence of these phases. The phase where the opinion of the society is overturned does not require the presence of agents with special characteristics, (stubborn, extremists, etc.); it results from the interplay of agents which have agreed on an extreme opinion with the remaining group that holds the opposite one. Among the former, some may be prone to compromise with other agents which are out of the majority group, these agents, according to their location in the network, may act like bridges between the two groups and slowly attract the whole society to the other extreme.

Bounded confidence opinion dynamics: A survey

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.

Mass media impact on opinion evolution in biased digital environments: a bounded confidence model

People increasingly shape their opinions by accessing and discussing content shared on social networking websites. These platforms contain a mixture of other users’ shared opinions and content from mainstream media sources. While online social networks have fostered information access and diffusion, they also represent optimal environments for the proliferation of polluted information and contents, which are argued to be among the co-causes of polarization/radicalization phenomena. Moreover, recommendation algorithms - intended to enhance platform usage - likely augment such phenomena, generating the so-called Algorithmic Bias. In this work, we study the effects of the combination of social influence and mass media influence on the dynamics of opinion evolution in a biased online environment, using a recent bounded confidence opinion dynamics model with algorithmic bias as a baseline and adding the possibility to interact with one or more media outlets, modeled as stubborn agents. We analyzed four different media landscapes and found that an open-minded population is more easily manipulated by external propaganda - moderate or extremist - while remaining undecided in a more balanced information environment. By reinforcing users’ biases, recommender systems appear to help avoid the complete manipulation of the population by external propaganda.

How Does Persuasion Happen? A Novel Bounded Confidence Opinion Dynamics Model Based on Social Judgment Theory

Persuasion exists in every aspect of social life. It is important to understand how persuasion works and how strong it is. In this article, we improve the classic Hegselmann–Krause model, one of the most famous bounded confidence models, and propose a novel opinion dynamics model to explain the process by which persuasion occurs from a systematic perspective. In our model, the concepts of latitudes of acceptance, noncommitment, and rejection from social judgment theory and the cognitive error in the process of persuasion, namely assimilation, are introduced. When people are exchanging their opinions with their neighbors, the opinions in the latitude of acceptance will be assimilated, those in the latitude of noncommitment will keep unchanged, and those in the latitude of rejection will not be considered. Theoretical proofs show that our model will converge to a stable state in a finite time. Numerical results of extensive simulation experiments on four datasets show the performance of the model. Furthermore, real social platform data and global COVID-19 vaccination data are analyzed to verify the effectiveness of the model in the decision-making process.

On The Role of Community Structure in Evolution of Opinion Formation: A New Bounded Confidence Opinion Dynamics

The process of opinion formation and evolution is an important part of the study of opinion dynamics. However, the existing opinion dynamics ignore the relationship between opinion evolution and community structure. To address the limitation, we build a novel bounded confidence opinion dynamics model (called LPA-HK for simplicity). Based on the Hegselmann-Krause (HK) model, the label propagation algorithm (LPA) is introduced in the LPA-HK model to study the relationship between opinion evolution and community partition from the micro-perspective of the evolutionary process. Each agent has an opinion as well as a label indicating the agent’s community. The iterations of agents’ states involve two stages: label and opinion updates. The labels and opinions of agents evolve dynamically until the group reaches a stable state. We perform simulations to examine the effect of the model parameters on opinion evolution and community partitions. We show that group reaches a consensus as the confidence level increases to a critical value and the partitions of high quality are obtained when the confidence level is greater than the critical value. We apply the LPA-HK model to three kinds of real networks to solve the community detection problem. The effectiveness of the LPA-HK model is verified by comparing the results with two existing community detection results based on modularity optimization algorithms.

Heterogeneous Opinion Dynamics With Confidence Thresholds Adaptation

Heterogeneous bounded confidence opinion dynamics with an adaptation policy of the agents’ confidence thresholds is considered. The strategy implemented by each agent consists of increasing his or her confidence thresholds when he or she has no active neighbor, thus strengthening his or her heterophilous propensity. Moreover, his or her adaptation algorithm is stopped if there is a minimal number of agents within his or her similarity interval. The heterogeneity of the model could lead to an interesting scenarios, e.g., steady-state oscillatory behaviors and practical clustering. Sufficient conditions for reaching in finite time a practical consensus are proposed. An upper bound on the maximum number of steady-state practical clusters, which depends on the parameters of the confidence thresholds adaptation algorithm, is obtained. The steady-state behaviors in the presence of one-sided confidence thresholds and stubbornness are discussed. Numerical simulations verify the theoretical results.

Finite-time convergence of opinion dynamics in homogeneous asymmetric bounded confidence 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.

Higher order interactions destroy phase transitions in Deffuant opinion dynamics model

Most opinion dynamics models are based on pairwise interactions. However in many real situations, discussions take place within groups of people. Here, we define a higher order Deffuant model by generalizing the original pairwise interaction model for bounded-confidence opinion-dynamics to interactions involving a group of agents of size k. The generalized model is naturally encoded in a hypergraph. We study this dynamics in different hypergraph topologies, from random hypergraph ensembles, to spatially embedded hyper-lattices. We show that including higher order interactions induces a drastic change in the onset of consensus for random hypergraphs; instead of the sharp phase transition, characteristic of the dyadic Deffuant model, the system undergoes a smooth size independent crossover to consensus, as the confidence value increases. This phenomenon is absent from regular hypergraphs, which conserve a phase transition.

Bounded Confidence Opinion Dynamics with Asch-Like Social Conformity in Paradigmatic and Real Networks

Bounded Confidence Opinion Dynamics with Asch-Like Social Conformity in Paradigmatic and Real Networks

Quasi-synchronization of bounded confidence opinion dynamics with a stochastic asynchronous rule

Quasi-synchronization of bounded confidence opinion dynamics with a stochastic asynchronous rule

Practical consensus in bounded confidence opinion dynamics

Opinion dynamics expressed by the bounded confidence discrete-time heterogeneous Hegselmann–Krause model is considered. A policy for the adaptation of the agents confidence thresholds based on heterophily, maximum number of neighbors and non-influencing similarity interval is proposed. The policy leads to the introduction of the concepts of practical clustering and practical consensus. Several properties of the agents dynamic behaviors are proved by exploiting the roles of the agents having at each time-step the maximum and the minimum opinions. The convergence in finite time to (a maximum number of) practical clusters and, for sufficiently large threshold bounds, the convergence to a practical consensus are proved. Sufficient conditions for reaching a practical consensus around a stubborn are derived too. Numerical simulations verify the theoretical results.

When open mindedness hinders consensus

We perform a detailed study of the Hegselmann-Krause bounded confidence opinion dynamics model with heterogeneousconfidence εi drawn from uniform distributions in different intervals [ε1, εu]. The phase diagram reveals a highly complex andnon-monotonous behaviour, with a re-entrant consensus phase in the region where fragmentation into multiple distinct opinionsis expected for the homogeneous case. A careful exploration of the phase diagram, along with an extensive finite-size analysis,allows us to identify the mechanism leading to this counter-intuitive behaviour. This systematic study over system sizes whichgo well beyond those of previous works, is enabled by an efficient algorithm presented in this article.

Noise-based synchronization of bounded confidence opinion dynamics in heterogeneous time-varying communication networks

A fundamental question that arises in the analysis of the noise-based synchronization of opinion dynamics with bounded confidence (BC) is what opinion structures can be synchronized by noise. In the standard Hegselmann-Krause (HK) model, each agent examines the opinion values of all other agents and then selects neighbors who are appropriate for opinion updating in accordance with the BC scheme. In reality, however, people are more likely to exchange opinions with only a limited set of individuals, resulting in a predetermined local communication network as postulated in the DeGroot model. In this paper, we develop a new model of opinion formation that endows the heterogeneous HK dynamics with a time-varying communication topology, and we investigate its noise-induced synchronization both mathematically and numerically. We show that opinion dynamics in this model are noise-synchronizable if and only if the switching communication graph is uniformly jointly connected. Our rigorous mathematical analysis and simulation experiments demonstrate that connected discourse topology in combination with fair amounts of noise can lead to a quasi-consensus that is inducible from any initial state or any underlying system size. Our findings have profound implications for the design of future strategies of social control and global opinion optimization.

Macroscopic Noisy Bounded Confidence Models With Distributed Radical Opinions

In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation. In particular, we, first, show the well-posedness of the dynamic equation, second, provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and, third, establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in second and third readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals.

Opinion consensus under external influences

As a means to regulate the continuous-time bounded confidence opinion dynamics, an exo-system to the original Hegselmann–Krause model is added. Some analysis is made about the properties of the combined system. Two theorems are provided in this article in terms of sufficient conditions of the exo-system that can guarantee opinion consensus for any initial conditions. Two more corollaries are given to describe the resulting synchronized opinions.

Signed bounded confidence models for opinion dynamics

The aim of this paper is to modify continuous-time bounded confidence opinion dynamics models so that “changes of opinion” (intended as changes of the sign of the initial states) are never induced during the evolution. Such sign invariance can be achieved by letting opinions of different sign localized near the origin interact negatively, or neglect each other, or even repel each other. In all cases, it is possible to obtain sign-preserving bounded confidence models with state-dependent connectivity and with a clustering behavior similar to that of a standard bounded confidence model.

Pairwise Stochastic Bounded Confidence Opinion Dynamics: Heavy Tails and Stability

Unlike traditional graph-based linear dynamics, where agents exchange opinions with their neighbors in a static social graph regardless of their differences in opinions, the bounded confidence opinion dynamics models exchange between agents with similar opinions. We generalize the bounded confidence opinion dynamics model by incorporating pairwise stochastic interactions, probabilistic influencing based on opinion differences and the self or endogenous evolutions of the agent opinions, which are represented by random processes. The opinion exchanges resulting from influencing have pairwise contraction effects, whereas endogenous motions have an expansive effect, for instance, of a diffusive nature. We analytically characterize the conditions under which this stochastic dynamics is stable in an appropriate sense. In the diffusive case, the presence of heavy tailed influence functions with a Pareto exponent of 2 is critical for stability (for a pair of agents an influence function maps opinion differences to probabilities of influence). Moreover, this model sheds light on dynamics that combine aspects of graph-based and bounded confidence dynamics.

Bounded confidence opinion dynamics with opinion leaders and environmental noises

Opinion dynamics is a kind of collective decision-making process and focuses on the study of evolution and formation of opinions within a human society. Particularly, bounded confidence rule is one of intrinsic interaction principles in the opinion dynamics. In this paper, a leader–follower opinion dynamics model is built, with the help of the bounded confidence rule, to consider the opinion formation of a community, which is constituted of opinion leaders and opinion followers. At the same time, environmental uncertainties are considered in the opinion formation and called as environmental noises, which are modeled as Gaussian stochastic processes. All the agents are assumed to have heterogeneous confidence levels. Then the impacts of the opinion leaders and the environmental noises on the final opinions of the opinion followers are analyzed. Finally, some simulation results are presented to demonstrate the collective opinion evolution in three cases: no opinion leader, single opinion leader and multiple opinion leaders.

Bounded Confidence Opinion Dynamics in a Social Network of Bayesian Decision Makers

Bounded confidence opinion dynamics model the propagation of information in social networks. However, in the existing literature, opinions are only viewed as abstract quantities without semantics rather than as part of a decision-making system. In this work, opinion dynamics are examined when agents are Bayesian decision makers that perform hypothesis testing or signal detection, and the dynamics are applied to prior probabilities of hypotheses. Bounded confidence is defined on prior probabilities through Bayes risk error divergence, the appropriate measure between priors in hypothesis testing. This definition contrasts with the measure used between opinions in standard models: absolute error. It is shown that the rapid convergence of prior probabilities to a small number of limiting values is similar to that seen in the standard Krause-Hegselmann model. The most interesting finding in this work is that the number of these limiting values and the time to convergence changes with the signal-to-noise ratio in the detection task. The number of final values or clusters is maximal at intermediate signal-to-noise ratios, suggesting that the most contentious issues lead to the largest number of factions. It is at these same intermediate signal-to-noise ratios at which the degradation in detection performance of the aggregate vote of the decision makers is greatest in comparison to the Bayes optimal detection performance. Real-world data from the United States Senate is examined in connection with the proposed model.