Models Of Continuous Opinion Dynamics Research Articles

Nonlinear Opinion Dynamics Models With Stubborn Individuals Over Signed Graphs

In this paper, we use nonlinear continuous opinion dynamics models to investigate how the opinions in social networks are affected by the existence of stubborn individuals who remain attached to their initial opinions. Specifically, we are interested in the case where trust-mistrust relations also exist in the network. We extend the existing models, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn positives</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn extremists</i> , by using signed graphs and allowing negative edges to model mistrust interactions. The effect of stubborn individuals is studied for both structurally balanced and unbalanced graphs. We show that, due to the stubborn individuals, a bipartite consensus (polarization) might not occur even if the associated graph is structurally balanced. Moreover, for structurally unbalanced graphs, the opinion vector does not always converge to 0 as compared to the well-known Altafini's model because of stubborn individuals. We present theorems that provide the above convergence guarantees.

Consensus, polarization, and coexistence in a continuous opinion dynamics model with quenched disorder.

A model of opinion dynamics is introduced in which each individual's opinion is measured on a bounded continuous spectrum. Each opinion is influenced heterogeneously by every other opinion in the population. It is demonstrated that consensus, polarization and a spread of moderate opinions are all possible within this model. Using dynamic mean-field theory, we are able to identify the statistical features of the interactions between individuals that give rise to each of the aforementioned emergent phenomena. The nature of the transitions between each of the observed macroscopic states is also studied. It is demonstrated that heterogeneity of interactions between individuals can lead to polarization, that mostly antagonistic or contrarian interactions can promote consensus at a moderate opinion, and that mostly reinforcing interactions encourage the majority to take an extreme opinion.

Finite-size effects on the convergence time in continuous-opinion dynamics

We study finite-size effects on the convergence time in a continuous-opinion dynamics model. In the model, each individual's opinion is represented by a real number on a finite interval, e.g., [0,1], and a uniformly randomly chosen individual updates its opinion by partially mimicking the opinion of a uniformly randomly chosen neighbor. We numerically find that the characteristic time to the convergence increases as the system size increases according to a particular functional form in the case of lattice networks. In contrast, unless the individuals perfectly copy the opinion of their neighbors in each opinion updating, the convergence time is approximately independent of the system size in the case of regular random graphs, uncorrelated scale-free networks, and complete graphs. We also provide a mean-field analysis of the model to understand the case of the complete graph.

Coupling Epidemiological Models with Social Dynamics.

In this work we study a Susceptible-Infected-Susceptible model coupled with a continuous opinion dynamics model. We assume that each individual can take measures to reduce the probability of contagion, and the level of effort each agent applies can change due to social interactions. We propose simple rules to model the propagation of behaviors that modify the level of effort, and analyze their impact on the dynamics of the disease. We derive a two dimensional set of ordinary differential equations describing the dynamic of the proportion of the number of infected individuals and the mean value of the effort parameter, and analyze the equilibria of the system. The stability of the endemic phase and disease free equilibria depends only on the mean value of the levels of efforts, and not on the initial distribution of levels of effort.

Approach to consensus in models of continuous-opinion dynamics: A study inspired by the physics of granular gases

A model for continuous-opinion dynamics is proposed and studied by taking advantage of its similarities with a mono-dimensional granular gas. Agents interact as in the Deffuant model, with a parameter α controlling the persuasibility of the individuals. The interaction coincides with the collision rule of two grains moving on a line, provided opinions and velocities are identified, with α being the so-called coefficient of normal restitution. Starting from the master equation of the probability density of all opinions, general conditions are given for the system to reach consensus. The case when the interaction frequency is proportional to the β-power of the relative opinions is studied in more detail. It is shown that the mean-field approximation to the master equation leads to the Boltzmann kinetic equation for the opinion distribution. In this case, the system always approaches consensus, which can be seen as the approach to zero of the opinion temperature, a measure of the width of the opinion distribution. Moreover, the long-time behaviour of the system is characterized by a scaling solution to the Boltzmann equation in which all time dependence occurs through the temperature. The case β=0 is related to the Deffuant model and is analytically soluble. The scaling distribution is unimodal and independent of α. For β>0 the distribution of opinions is unimodal below a critical value of |α|, being multimodal with two maxima above it. This means that agents may approach consensus while being polarized. Near the critical points and for |α|≥0.4, the distribution of opinions is well approximated by the sum of two Gaussian distributions. Monte Carlo simulations are in agreement with the theoretical results.

Continuous Opinion Dynamics with Anti-conformity Behavior

We propose an appropriate updating rule of continuous opinions for modeling anti-conformity behavior, defined according to the repelling function, which gives the shift of the opinion based on the current opinion and the reference opinion for an agent. Two models of continuous opinion dynamics (with opinion value on a continuous scale [0,1]) are studied in undirected networks, by introducing the heterogeneity in the sense of conformity and anti-conformity behavior either in nodes or in links. In the first one, the society is composed of both conformist and anti-conformist agents. Conformist agents update their opinions following the DeGroot rule with equal weights. However, anti-conformist agents would like to repel from others, and the repelling level is negatively related to the opinion distance between the anti-conformist and her reference point. No consensus will be reached for any connected network in the presence of anti-conformist agents. Instead, opinions converge to a disagreement or oscillate over time. In the second part, by supposing a signed graph where agents have positive links (+1) with their friends and negative links (-1) with their enemies, agents update their opinion as the sum of the averaged opinion of their friends and repelling value from their enemies. When the network is balanced, i.e., there are two communitarian groups, and each sub-network corresponding to each group is connected and the initial opinion ranges of the two groups are disjoint, the consensus within each group is guaranteed. Both synchronous and asynchronous updating models are discussed in these two parts.

Emergence of moderate opinions as a consequence of group pressure

In this work, we study a continuous opinion dynamics model considering 3-agent interactions and group pressure. Agents interact in a fully-connected population, and two parameters govern the dynamics: the agents’ convictions [Formula: see text] that are homogeneous in the population, and the group pressure [Formula: see text]. Stochastic parameters also drive the interactions. Our analytical and numerical results indicate that the model undergoes symmetry-breaking transitions at distinct critical points [Formula: see text] for any value of [Formula: see text], i.e. the transition can be suppressed for sufficiently high group pressure. Such transition separates two phases: for any [Formula: see text], the order parameter [Formula: see text] is identically null ([Formula: see text], a symmetric, absorbing phase), while for [Formula: see text], we have [Formula: see text], i.e. a symmetry-broken phase (ferromagnetic). The numerical simulations also reveal that the increase of group pressure leads to a wider distribution of opinions, decreasing the extremism in the population.

Impact of temporal network structures on the speed of consensus formation in opinion dynamics

Opinion dynamics on networks has wide applications to empirical and engineered systems and profound prospects in the general study of complex systems. Many efforts have been devoted to understanding how opinion dynamics is affected by network topology. However, social interactions are best characterized as temporal networks in which ordering of interactions cannot be ignored. Temporal activity patterns including heterogeneous contact strength and interevent times, turnover edge/node dynamics, and daily patterns could have significant effects that would not be captured by static aggregate network representations. In this paper, we study the effects of such temporal patterns on the speed of consensus formation in various models of continuous opinion dynamics using four empirical data sets, including three human face-to-face networks from different real-world settings and a social insect interaction network. We find that static, aggregated networks consistently overestimate the speed of simulated consensus formation while weight heterogeneity associated with frequency of interactions has an inhibitory effect on consensus formation relative to the behavior on unweighted networks. Moreover, the speed of consensus formation is found to be highly sensitive to nodal lifetimes, suggesting that randomization protocols that dramatically alter the distribution of lifetimes cannot be relied upon as reference models. On the other hand, temporal patterns including burstiness of interevent times and the lifetime of edges are found to have insignificant effects on consensus formation. The sensitivity to nodal lifetimes is further demonstrated via synthetic networks generated by an activity-driven model with tunable nodal lifetimes.

Algorithmic bias amplifies opinion fragmentation and polarization: A bounded confidence model.

The flow of information reaching us via the online media platforms is optimized not by the information content or relevance but by popularity and proximity to the target. This is typically performed in order to maximise platform usage. As a side effect, this introduces an algorithmic bias that is believed to enhance fragmentation and polarization of the societal debate. To study this phenomenon, we modify the well-known continuous opinion dynamics model of bounded confidence in order to account for the algorithmic bias and investigate its consequences. In the simplest version of the original model the pairs of discussion participants are chosen at random and their opinions get closer to each other if they are within a fixed tolerance level. We modify the selection rule of the discussion partners: there is an enhanced probability to choose individuals whose opinions are already close to each other, thus mimicking the behavior of online media which suggest interaction with similar peers. As a result we observe: a) an increased tendency towards opinion fragmentation, which emerges also in conditions where the original model would predict consensus, b) increased polarisation of opinions and c) a dramatic slowing down of the speed at which the convergence at the asymptotic state is reached, which makes the system highly unstable. Fragmentation and polarization are augmented by a fragmented initial population.

Nonequilibrium opinion dynamics on triangular, honeycomb, and Kagome lattices

The Ising model on all Archimedean lattices exhibits spontaneous ordering. Three examples of these lattices, namely triangular ([Formula: see text]), honeycomb [Formula: see text] and Kagome [Formula: see text] lattices, are considered to study the kinetic continuous opinion dynamics model (KCOD) through extensive Monte Carlo simulations. The order/disorder phase transition is observed in all lattices for the KCOD. The estimated values of the critical disorder parameter are [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] lattices, respectively. The critical exponents [Formula: see text], [Formula: see text] and [Formula: see text] for the model are [Formula: see text], [Formula: see text], and [Formula: see text]; [Formula: see text], [Formula: see text], and [Formula: see text]; [Formula: see text], [Formula: see text], and [Formula: see text], for [Formula: see text], [Formula: see text] and [Formula: see text] lattices, respectively. These results agree with the majority-vote model on ([Formula: see text]), ([Formula: see text]), and [Formula: see text] lattices but are different from KCOD model results on square lattices [Formula: see text].

Kinetic Continuous Opinion Dynamics Model on Two Types of Archimedean Lattices

Here, the critical properties of kinetic continuous opinion dynamics model are studied on ($4,6,12$) and ($4,8^2$) Archimedean lattices. We obtain $p_c$ and the critical exponents from Monte Carlo simulations and finite size scaling. We found out the values of the critical points and Binder cumulant that are $p_c=0.086(3)$ and $O_4^*=0.59(2)$ for ($4,6,12$); and $p_c=0.109(3)$ and $O_4^*=0.606(5)$ for ($4,8^2$) lattices and also the exponent ratios $\beta/\nu$, $\gamma/\nu$ and $1/\nu$ are respectively: $0.23(7)$, $1.43(5)$ and $ 0.60(3)$ for ($4,6,12$); and $0.149(4)$, $1.56(4)$ and $0.94(4)$ for ($4,8^2$) lattices.

The KCOD Model on (3,4,6,4) and (34,6) Archimedean Lattices

Through Monte Carlo simulations, we studied the critical properties of kinetic models of continuous opinion dynamics on ( 3 , 4 , 6 , 4 ) and ( 3 4 , 6 ) Archimedean lattices. We obtain p c and the critical exponents’ ratio from extensive Monte Carlo studies and finite size scaling. The calculated values of the critical points and Binder cumulant are p c = 0 . 085 ( 6 ) and O 4 * = 0 . 605 ( 9 ) ; and p c = 0 . 146 ( 5 ) and O 4 * = 0 . 606 ( 3 ) for ( 3 , 4 , 6 , 4 ) and ( 3 4 , 6 ) lattices, respectively, while the exponent ratios β / ν , γ / ν and 1 / ν are, respectively: 0 . 126 ( 1 ) , 1 . 50 ( 7 ) , and 0 . 90 ( 5 ) for ( 3 , 4 , 6 , 4 ); and 0 . 125 ( 3 ) , 1 . 54 ( 6 ) , and 0 . 99 ( 3 ) for ( 3 4 , 6 ) lattices. Our new results agree with majority-vote model on previously studied regular lattices and disagree with the Ising model on square-lattice.

Innovation governs everything eventually: Extensions of the DeGroot model

We study the DeGroot model for continuous opinion dynamics under the influence of innovations. In the original model, individuals’ opinions, after given their initial values, evolve merely according to the given learning topology. The main contribution of this paper is that external innovation effects are introduced: each individual is given the opportunity to change her opinion to a randomly selected opinion according to a given distribution on the opinion space and then the external opinion is either adapted by the individual, or combined into her learning process. It turns out that all the classical results of the DeGroot model are violated in this new model. We prove that convergence can still be guaranteed in the expectation sense, regardless of the learning topology. We also study the steady distributions of opinions among the society and the time spent to reach a steady state by means of Monte-Carlo simulations.

A Novel Private Attitude and Public Opinion Dynamics Model for Simulating Pluralistic Ignorance and Minority Influence

Pluralistic ignorance, a well-documented socio-psychological conformity phenomenon, involves discrepancies between private attitude and public opinion in certain social contexts. However, continuous opinion dynamics models based on a bounded confidence assumption fail to accurately model pluralistic ignorance because they do not address scenarios in which non-conformists do not need to worry about holding and expressing conflicting opinions. Such scenarios reduce the power of continuous opinion dynamics models to explain why certain groups doubt or change their opinions in response to minority views. To simulate the effects of (a) private acceptance of informational social influence and (b) public compliance with normative social influence on pluralistic ignorance and minority influences, we have created an agent-based simulation model in which attitude and opinion respectively represent an agent's private and expressed thoughts. Results from a series of simulation experiments indicate model validity equal to or exceeding those of existing opinion dynamics models that are also based on the bounded confidence assumption, but with different dynamics and outcomes in terms of collective opinion and attitude. The results also support the use of our proposed model for computational social psychology applications.

Modeling self-perception agents in an opinion dynamics propagation society

In previous continuous opinion dynamics models based on bounded confidence assumptions, individuals can only influence each other’s opinions when those opinions are sufficiently close; subsequently, agents construct their self-opinions using opinions collected from other agents. All of these models lack the element of an agent’s inner self-attitude. In this paper we describe our proposal for a self-perception model in which agents are aware of differences between their attitudes and expressed opinions on specific issues. Our agents are based on a mix of self-perception and cognitive dissonance theory that allows them to self-adjust discomfort caused by inconsistencies between inner attitudes and expressed opinions. Results from a series of simulation experiments indicate that our model captures the gap between inner attitude and external opinion in explaining the private acceptance/public conformity phenomenon. We conclude with a demonstration of how our proposed model can be used in sociological studies of pluralistic ignorance.

The noisy Hegselmann-Krause model for opinion dynamics

In the model for continuous opinion dynamics introduced by Hegselmann and Krause, each individual moves to the average opinion of all individuals within an area of confidence. In this work we study the effects of noise in this system. With certain probability, individuals are given the opportunity to change spontaneously their opinion to another one selected randomly inside the opinion space with different rules. If the random jump does not occur, individuals interact through the Hegselmann-Krause's rule. We analyze two cases, one where individuals can carry out opinion random jumps inside the whole opinion space, and other where they are allowed to perform jumps just inside a small interval centered around the current opinion. We found that these opinion random jumps change the model behavior inducing interesting phenomena. Using pattern formation techniques, we obtain approximate analytical results for critical conditions of opinion cluster formation. Finally, we compare the results of this work with the noisy version of the Deffuant et al. model for continuous-opinion dynamics.

Multi-Level Opinion Dynamics under Bounded Confidence

Opinion dynamics focuses on the opinion evolution in a social community. Recently, some models of continuous opinion dynamics under bounded confidence were proposed by Deffuant and Krause, et al. In the literature, agents were generally assumed to have a homogeneous confidence level. This paper proposes an extended model for a group of agents with heterogeneous confidence levels. First, a social differentiation theory is introduced and a social group is divided into opinion subgroups with distinct confidence levels. Second, a multi-level heterogeneous opinion formation model is formulated under the framework of bounded confidence. Finally, computer simulations are conducted to study the collective opinion evolution, focusing on three key factors: the fractions of heterogeneous agents, the initial opinions, and the group size. The simulation results demonstrate that the number of final opinions depends on the fraction of close-minded agents when the group size and the initial opinions are fixed; the final opinions converge more easily when the initial opinions are closer; and the number of final opinions can be approximately modeled by a linear increasing function of the group size and the increasing rate is the fraction of close-minded agents.

Asymptotic analysis of continuous opinion dynamics models under bounded confidence

This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded confidence of Deffuant-Weisbuch type.Focusing on the Cauchy Problem related to compromise models with homogeneous bound of confidence, a general well-posedness result is provided and a systematic study of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.

Collective opinion and attitude dynamics dependency on informational and normative social influences

In a continuous opinion dynamics model using a bounded confidence assumption, individuals can only influence each other’s opinions when those opinions are sufficiently close. However, we often observe real-world cases in which opinions are very different, yet individuals feel compelled to change their ideas to conform with their peers or superiors (or in rare cases, are willing to change them voluntarily). In other words, individuals tend to consider the practical value of conformity and worry about rejection if they do not adopt the opinions of the majority. To explore the influences of private acceptance of informational social influences and public compliance with normative social influences on collective opinion and attitude dynamics, we have created a model in which attitude and opinion respectively represent an agent’s private and expressed thoughts. Results from a series of simulation experiments indicate that our simplified model is as valid as previous opinion dynamics models also based on the bounded confidence assumption, but with different dynamics and outcomes regarding group opinion and attitude. To demonstrate our proposed model’s potential value and applications, we briefly discuss two issues of import to sociologists: pluralistic ignorance formation and destruction and minority influence.

Diffusing opinions in bounded confidence processes

We study the effects of diffusing opinions on the Deffuant et al. model for continuous opinion dynamics. Individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside an interval centered around the present opinion. We show that diffusion induces an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion clusters are formed, although with some opinion spread inside them. If the diffusion jumps are not large, clusters coalesce, so that weak diffusion favors opinion consensus. A master equation for the process described above is presented. We find that the master equation and the Monte Carlo simulations do not always agree due to finite-size induced fluctuations. Using a linear stability analysis we can derive approximate conditions for the transition between opinion clusters and the disordered state. The linear stability analysis is compared with Monte Carlo simulations. Novel interesting phenomena are analyzed.