We propose a model of continuous opinion dynamics, where mutual interactions can be both positive and negative. Different types of distributions for the interactions, all characterized by a single parameter $p$ denoting the fraction of negative interactions, are considered. Results from exact calculation of a discrete version and numerical simulations of the continuous version of the model indicate the existence of a universal continuous phase transition at p=p_c below which a consensus is reached. Although the order-disorder transition is analogous to a ferromagnetic-paramagnetic phase transition with comparable critical exponents, the model is characterized by some distinctive features relevant to a social system.
In this work we study the critical behavior of a three-state ($+1$, $-1$, $0$) opinion model with independence. Each agent has a probability $q$ to act as independent, i.e., he/she can choose his/her opinion independently of the opinions of the other agents. On the other hand, with the complementary probability $1-q$ the agent interacts with a randomly chosen individual through a kinetic exchange. Our analytical and numerical results show that the independence mechanism acts as a noise that induces an order-disorder transition at critical points $q_{c}$ that depend on the individuals' flexibil…
We present here the exact solution of an infinite range, discrete, opinion formation model. The model shows an active-absorbing phase transition, similar to that numerically found in its recently proposed continuous version (Lallouache et al., Phys. Rev E 82, 056112 (2010)). Apart from the two-agent interactions here we also report the effect of having three agent interactions. The phase diagram has a continuous transition line (two agent interaction dominated) and a discontinuous transition line (three agent interaction dominated) separated by a tricritical point.
We propose a model of binary opinion in which the opinion of the individuals change according to the state of their neighbouring domains. If the neighbouring domains have opposite opinions, then the opinion of the domain with the larger size is followed. Starting from a random configuration, the system evolves to a homogeneous state. The dynamical evolution show novel scaling behaviour with the persistence exponent $θ\simeq 0.235$ and dynamic exponent $z \simeq1.02 \pm 0.02$. Introducing disorder through a parameter called rigidity coefficient $ρ$ (probability that people are completely rigid …
The concept of a bounded confidence level is incorporated in a nonconservative kinetic exchange model of opinion dynamics model where opinions have continuous values $\in [-1,1]$. The characteristics of the unrestricted model, which has one parameter $λ$ representing conviction, undergo drastic changes with the introduction of bounded confidence parametrised by $δ$. Three distinct regions are identified in the phase diagram in the $δ-λ$ plane and the evidences of a first order phase transition for $δ\geq 0.3$ are presented. A neutral state with all opinions equal to zero occurs for $λ\leq λ_{c…
In this work we tackle a kinetic-like model of opinions dynamics in a networked population endued with a quenched plurality and polarization. Additionally, we consider pairwise interactions that are restrictive, which is modeled with a smooth bounded confidence. Our results show the interesting emergence of nonequilibrium hysteresis and heterogeneity-assisted ordering. Such counterintuitive phenomena are robust to different types of network architectures such as random, small-world and scale-free.
We propose a minimal model for the collective dynamics of opinion formation in the society, by modifying kinetic exchange dynamics studied in the context of income, money or wealth distributions in a society. This model has an intriguing spontaneous symmetry breaking transition.
We review in details some recently proposed kinetic models of opinion dynamics. We discuss the several variants including a generalised model. We provide mean field estimates for the critical points, which are numerically supported with reasonable accuracy. Using non-equilibrium relaxation techniques, we also investigate the nature of phase transitions observed in these models. We study the nature of correlations as the critical points are approached, and comment on the universality of the phase transitions observed.
We analyze the critical behavior of a class of discrete opinion models in the presence of disorder. Within this class, each agent opinion takes a discrete value ($\pm 1$ or 0) and its time evolution is ruled by two terms, one representing agent-agent interactions and the other the degree of conviction or persuasion (a self-interaction). The mean-field limit, where each agent can interact evenly with any other, is considered. Disorder is introduced in the strength of both interactions, with either quenched or annealed random variables. With probability $p$ (1-$p$), a pairwise interaction reflec…
We study a model of continuous opinion dynamics with both positive and negative mutual interaction. The model shows a continuous phase transition between a phase with consensus (order) and a phase having no consensus (disorder). The mean field version of the model was already studied. Using extensive numerical simulations, we study the same model in $2$ and $3$ dimensions. The critical points of the phase transitions for various cases and the associated critical exponents have been estimated. The universality class of the phase transitions in the model is found to be same as Ising model in the…
In this work we study opinion formation on a fully-connected population participating of a public debate with two distinct choices, where the agents may adopt three different attitudes (favorable to either one choice or to the other, or undecided). The interactions between agents occur by pairs and are competitive, with couplings that are either negative with probability $p$ or positive with probability $1-p$. This bimodal probability distribution of couplings produces a behavior similar to the one resulting from the introduction of Galam's contrarians in the population. In addition, we consid…
We show using scaling arguments and Monte Carlo simulations that a class of binary interacting models of opinion evolution belong to the Ising universality class in presence of an annealed noise term of finite amplitude. While the zero noise limit is known to show an active-absorbing transition, addition of annealed noise induces a continuous order-disorder transition with Ising universality class in the infinite-range (mean field) limit of the models.
We propose a minimal multi-agent model for the collective dynamics of opinion formation in the society, by modifying kinetic exchange dynamics studied in the context of income, money or wealth distributions in a society. This model has an intriguing spontaneous symmetry breaking transition to polarized opinion state starting from non-polarized opinion state. In order to analyze the model, we introduce an iterative map version of the model, which has very similar statistical characteristics. An approximate theoretical analysis of the numerical results are also given, based on the iterative map …
We investigate opinion formation in a kinetic exchange opinion model, where opinions are represented by numbers in the real interval $[-1,1]$ and agents are typified by the individual degree of conviction about the opinion that they support. Opinions evolve through pairwise interactions governed by competitive positive and negative couplings, that promote imitation and dissent, respectively. The model contemplates also another type of nonconformity such that agents can occasionally choose their opinions independently of the interactions with other agents. The steady states of the model as a fu…
We study the percolation transition of the geometrical clusters in the square lattice LCCC model (a kinetic opinion exchange model introduced by Lallouache et al. in Phys. Rev. E 82 056112 (2010)) with the change in conviction and influencing parameter. The cluster comprises of the adjacent sites having an opinion value greater than or equal to a prefixed threshold value of opinion (Ω). The transition point is different from that obtained for the transition of the order parameter (average opinion value) found by Lallouache et al. Although the transition point varies with the change in the thre…
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 th…
We propose an opinion dynamics model based on Latané's social impact theory. Actors in this model are heterogeneous and, in addition to opinions, are characterised by their varying levels of persuasion and support. The model is tested for two and three initial opinions randomly distributed among actors. We examine how the noise (randomness of behaviour) and the flow of information among actors affect the formation and spread of opinions. Our main research involves the process of opinion formation and finding phases of the system in terms of parameters describing noise and flow of the informati…
Inferring the microscopic mechanisms of opinion dynamics using a kinetic Ising model
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；LIP6（LIP6实验室）；Complex Systems Institute of Paris île-de-France (ISC-PIF, UAR3611)（巴黎-法兰西岛复杂系统研究所）；Centre d’Analyse et de Mathématique Sociales (CAMS, UMR8557)（社会分析与数学中心）；Université Paris-Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）
In this work we study the critical behavior of a three-state opinion model in the presence of noise. This noise represents the independent behavior, that plays the role of social temperature. Each agent on a regular D-dimensional lattice has a probability $q$ to act as independent, i.e., he can choose his opinion independent of the opinions of his neighbors. Furthermore, with the complementary probability $1-q$ the agent interacts with a randomly chosen nearest neighbor through a kinetic exchange. Our numerical results suggest that the model undergoes nonequilibrium phase transitions at critic…
Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani's kinetic opinion formation model and the leader-follower model of Düring et al., and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker-Planck-type systems. Numerical example…
We introduce and discuss kinetic models of opinion formation on social networks in which the distribution function depends on both the opinion and the connectivity of the agents. The opinion formation model is subsequently coupled with a kinetic model describing the spreading of popularity of a product on the web through a social network. Numerical experiments on the underlying kinetic models show a good qualitative agreement with some measured trends of hashtags on social media websites and illustrate how companies can take advantage of the network structure to obtain at best the advertisemen…
A general 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, polarisation 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 i…
We investigate the opinion formation among the peoples and multiple political parties using the one dimensional relativistic Boltzmann-Vlasov equation for multi-components. A political party is constituted of politicians. The opinion formation depends on self-thinkings of peoples and politicians, and the constraint of the political party over opinions of politicians, when we restrict ourselves to the conciliatory exchange of opinions between two individuals. In particular, shock like profiles are obtained in the distribution of opinions of peoples, when the self-thinking of politicians are abs…
We study the Deffuant et al. model for continuous--opinion dynamics under the influence of noise. In the original version of this model, individuals meet in random pairwise encounters after which they compromise or not depending of a confidence parameter. Free will is introduced in the form of noisy perturbations: individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside the whole opinion space. We derive the master equation of this process. One of the main effects of noise is to induce an order-disorder transition. In the di…
Biswas-Chatterjee-Sen (BChS) kinetic exchange opinion model on modular networks
School of Computational & Integrative Sciences, Jawaharlal Nehru University（计算与整合科学学院，贾瓦哈拉尔·尼赫鲁大学）；Department of Physics, SRM University - AP（物理系，SRM大学-安得拉邦分校）
In this work we study a 3-state opinion formation model considering two distinct mechanisms, namely independence and conviction. Independence is introduced in the model as a noise, by means of a probability of occurrence $q$. On the other hand, conviction acts as a disorder in the system, and it is introduced by two discrete probability distributions. We analyze the effects of such two mechanisms on the phase transitions of the model, and we found that the critical exponents are universal over the order-disorder frontier, presenting the same universality class of the mean-field Ising model. In…
We study an influence network of voters subjected to correlated disordered external perturbations, and solve the dynamical equations exactly for fully connected networks. The model has a critical phase transition between disordered unimodal and ordered bimodal distribution states, characterized by an increase in the vote-share variability of the equilibrium distributions. The random heterogeneities in the external perturbations are shown to affect the critical behavior of the network relative to networks without disorder. The size of the shift in the critical behavior essentially depends on th…
Recently, a model of opinion formation with kinetic exchanges has been proposed in which a spontaneous symmetry breaking transition was reported [M. Lallouache et al, Phys. Rev. E, {\bf 82} 056112 (2010)]. We generalise the model to incorporate two parameters, $λ$, to represent conviction and $μ$, to represent the influencing ability of individuals. A phase boundary given by $λ=1-μ/2$ is obtained separating the symmetric and symmetry broken phases: the effect of the influencing term enhances the possibility of reaching a consensus in the society. The time scale diverges near the phase boundary…
We study a recently proposed kinetic exchange opinion model (Lallouache et. al., Phys. Rev E 82:056112, 2010) in the limit of a single parameter map. Although it does not include the essentially complex behavior of the multiagent version, it provides us with the insight regarding the choice of order parameter for the system as well as some of its other dynamical properties. We also study the generalized two- parameter version of the model, and provide the exact phase diagram. The universal behavior along this phase boundary in terms of the suitably defined order parameter is seen.
In this work we study a model of opinion dynamics considering activation/deactivation of agents. In other words, individuals are not static and can become inactive and drop out from the discussion. A probability $w$ governs the deactivation dynamics, whereas social interactions are ruled by kinetic exchanges, considering competitive positive/negative interactions. Inactive agents can become active due to interactions with active agents. Our analytical and numerical results show the existence of two distinct nonequilibrium phase transitions, with the occurrence of three phases, namely ordered (…
Understanding the impact of collective social phenomena in epidemic dynamics is a crucial task to effectively contain the disease spread. In this work we build a mathematical description for assessing the interplay between opinion polarization and the evolution of a disease. The proposed kinetic approach describes the evolution of aggregate quantities characterizing the agents belonging to epidemiologically relevant states, and will show that the spread of the disease is closely related to consensus dynamics distribution in which opinion polarization may emerge. In the present modelling framew…
In this paper an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model we analyze a microscopic model of opinion formation under constraints. For this problem a Boltzmann-type equation based on a model predictive control formulation is introduced and discussed. In particular, the receding horizon strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions of stationary solutions are giv…
We consider a nonlinear kinetic equation of Boltzmann type which takes into account the influence of conviction during the formation of opinion in a system of agents which interact through the binary exchanges introduced in [G. Toscani, Commun. Math. Sci. 4, 481 (2006)]. The original exchange mechanism, which is based on the human tendency to compromise and change of opinion through self-thinking, is here modified in the parameters of the compromise and diffusion terms, which now are assumed to depend on the personal degree of conviction. The numerical simulations show that the presence of con…
In this work, we investigate interactions that simultaneously order a system locally, while keeping it globally disordered. The study is done in the context of the emergence of diversity in opinion propagation models with interactions rooted in conformity, but some suggestions on how this could be extended to other topics (like ecology and neuroscience) are also made. We do this by introducing a generic modification that can be added to different opinion propagation models (and other agent based models) and that seems to introduce a global tendency towards diversity without leading the system …
In this work we study a 3-state ($+1$, $-1$, $0$) opinion model in the presence of noise and disorder. We consider pairwise competitive interactions, with a fraction $p$ of those interactions being negative (disorder). Moreover, there is a noise $q$ that represents the probability of an individual spontaneously change his opinion to the neutral state. Our aim is to study how the increase/decrease of the fraction of neutral agents affects the critical behavior of the system and the evolution of opinions. We derive analytical expressions for the order parameter of the model, as well as for the s…
In this work we consider the presence of contrarian agents in discrete 3-state kinetic exchange opinion models. The contrarians are individuals that adopt the choice opposite to the prevailing choice of their contacts, whatever this choice is. We consider binary as well as three-agent interactions, with stochastic parameters, in a fully-connected population. Our numerical results suggest that the presence of contrarians destroys the absorbing state of the original model, changing the transition to the para-ferromagnetic type. In this case, the consequence for the society is that the three opin…
In this paper, we propose a Boltzmann-type kinetic description of opinion formation on social networks, which takes into account a general connectivity distribution of the individuals. We consider opinion exchange processes inspired by the Sznajd model and related simplifications but we do not assume that individuals interact on a regular lattice. Instead, we describe the structure of the social network statistically, assuming that the number of contacts of a given individual determines the probability that their opinion reaches and influences the opinion of another individual. From the kineti…
From graphons to real-world networks: kinetic opinion dynamics under selective media influence
Mathematics Institute, University of Warwick（华威大学数学研究所）；Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino（都灵理工大学“G.L.拉格朗日”数学科学系）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino（意大利国家核物理研究所（INFN）都灵分部）
A model for opinion dynamics (Model I) has been recently introduced in which the binary opinions of the individuals are determined according to the size of their neighboring domains (population having the same opinion). The coarsening dynamics of the equivalent Ising model shows power law behavior and has been found to belong to a new universality class with the dynamic exponent $z=1.0 \pm 0.01$ and persistence exponent $θ\simeq 0.235$ in one dimension. The critical behavior has been found to be robust for a large variety of annealed disorder that has been studied. Further, by mapping Model I …
