Impact of contrarians and intransigents in a kinetic model of opinion dynamics

Democratic Thwarting of Majority Rule in Opinion Dynamics: 1. Unavowed Prejudices Versus Contrarians

I study the conditions under which the democratic dynamics of a public debate drives a minority-to-majority transition. A landscape of the opinion dynamics is thus built using the Galam Majority Model (GMM) in a 3-dimensional parameter space for three different sizes, r=2,3,4, of local discussion groups. The related parameters are (p0,k,x), the respective proportions of initial agents supporting opinion A, unavowed tie prejudices breaking in favor of opinion A, and contrarians. Combining k and x yields unexpected and counterintuitive results. In most of the landscape the final outcome is predetermined, with a single-attractor dynamics, independent of the initial support for the competing opinions. Large domains of (k,x) values are found to lead an initial minority to turn into a majority democratically without any external influence. A new alternating regime is also unveiled in narrow ranges of extreme proportions of contrarians. The findings indicate that the expected democratic character of free opinion dynamics is indeed rarely satisfied. The actual values of (k,x) are found to be instrumental to predetermining the final winning opinion independently of p0. Therefore, the conflicting challenge for the predetermined opinion to lose is to modify these values appropriately to become the winner. However, developing a model which could help in manipulating public opinion raises ethical questions. This issue is discussed in the Conclusions.

Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias

We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability p(t) (majority behavior) or the opposite opinion with probability 1-p(t) (contrarian behavior). The probability of following the majority rule p(t) varies with the temperature T and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as T is varied. A transition temperature Tc separates a bimodal oscillatory regime for TTc in which m oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature T*, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that T* and Tc are closely related.

Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance

We study the contrarian voter model for opinion formation in a society under the influence of an external oscillating propaganda and stochastic noise. Each agent of the population can hold one of two possible opinions on a given issue—against or in favor—and interacts with its neighbors following either an imitation dynamics (voter behavior) or an anti-alignment dynamics (contrarian behavior): each agent adopts the opinion of a random neighbor with a time-dependent probability p(t), or takes the opposite opinion with probability 1−p(t). The imitation probability p(t) is controlled by the social temperature T, and varies in time according to a periodic field that mimics the influence of an external propaganda, so that a voter is more prone to adopt an opinion aligned with the field. We simulate the model in complete graph and in lattices, and find that the system exhibits a rich variety of behaviors as T is varied: opinion consensus for T=0, a bimodal behavior for T<Tc, an oscillatory behavior where the mean opinion oscillates in time with the field for T>Tc, and full disorder for T≫1. The transition temperature Tc vanishes with the population size N as Tc≃2/lnN in complete graph. In addition, the distribution of residence times tr in the bimodal phase decays approximately as tr−3/2. Within the oscillatory regime, we find a stochastic resonance-like phenomenon at a given temperature T*. Furthermore, mean-field analytical results show that the opinion oscillations reach a maximum amplitude at an intermediate temperature, and that exhibit a lag with respect to the field that decreases with T.

Opinion dynamics: public and private

We study here the dynamics of opinion formation in a society where we take into account the internally held beliefs and externally expressed opinions of the individuals, which are not necessarily the same at all times. While these two components can influence one another, their difference, both in dynamics and in the steady state, poses interesting scenarios in terms of the transition to consensus in the society and characterizations of such consensus. Here we study this public and private opinion dynamics and the critical behaviour of the consensus forming transitions, using a kinetic exchange model. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Double transition in kinetic exchange opinion models with activation dynamics

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 [Formula: see text] 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 non-equilibrium phase transitions, with the occurrence of three phases, namely ordered (ferromagnetic-like), disordered (paramagnetic-like) and absorbing phases. The absorbing phase represents a collective state where all agents are inactive, i.e. they do not participate in the dynamics, inducing a frozen state. We determine the critical value [Formula: see text] above which the system is in the absorbing phase independently of the other parameters. We also verify a distinct critical behaviour for the transitions among different phases. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Analytical approach to the Axelrod model based on similarity vectors

Complex problems of social interaction are usually studied within the framework of agent-based models. Some of these problems, such as issue alignment and opinion polarization, are better suited in the framework of n-dimensional opinion space. Although this kind of complex problem may be explored by numerical simulations, these simulations can hinder our ability to obtain general results. In this work, we show how, under certain conditions, a classical multidimensional opinion model such as the Axelrod model can give rise to a closed set of master equations in terms of vector similarities between agents. The analytical results fully agree with the simulations on complete networks, accurately predict the similarity distribution of the whole system in sparse topologies, and provide a good approximation of the similarity of physical links that improves when the mean degree of the system increases.

Polarizing crowds: Consensus and bipolarization in a persuasive arguments model

Understanding the opinion formation dynamics in social systems is of vast relevance in diverse aspects of society. In particular, it is relevant for political deliberation and other group decision-making processes. Although previous research has reported different approaches to model social dynamics, most of them focused on interaction mechanisms where individuals modify their opinions in line with the opinions of others, without invoking a latent mechanism of argumentation. In this paper, we present a model where changes of opinion are due to explicit exchanges of arguments, and we analyze the emerging collective states in terms of simple dynamic rules. We find that, when interactions are equiprobable and symmetrical, the model only shows consensus solutions. However, when either homophily, confirmation bias, or both are included, we observe the emergence and dominance of bipolarization, which appears due to the fact that individuals are not able to accept the contrary information from their opponents during exchanges of arguments. In all cases, the predominance of each stable state depends on the relation between the number of agents and the number of available arguments in the discussion. Overall, this paper describes the dynamics and shows the conditions wherein deliberative agents are expected to construct polarized societies.

Threshold q-voter model

We introduce the threshold q-voter opinion dynamics where an agent, facing a binary choice, can change its mind when at least q_{0} among q neighbors share the opposite opinion. Otherwise, the agent can still change its mind with a certain probability ɛ. This threshold dynamics contemplates the possibility of persuasion by an influence group even when there is not full agreement among its members. In fact, individuals can follow their peers not only when there is unanimity (q_{0}=q) in the lobby group, as assumed in the q-voter model, but also, depending on the circumstances, when there is simple majority (q_{0}>q/2), Byzantine consensus (q_{0}>2q/3), or any minimal number q_{0} among q. This realistic threshold gives place to emerging collective states and phase transitions which are not observed in the standard q voter. The threshold q_{0}, together with the stochasticity introduced by ɛ, yields a phenomenology that mimics as particular cases the q voter with stochastic drivings such as nonconformity and independence. In particular, nonconsensus majority states are possible, as well as mixed phases. Continuous and discontinuous phase transitions can occur, but also transitions from fluctuating phases into absorbing states.

Zealots in the mean-field noisy voter model

The influence of zealots on the noisy voter model is studied theoretically and numerically at the mean-field level. The noisy voter model is a modification of the voter model that includes a second mechanism for transitions between states: Apart from the original herding processes, voters may change their states because of an intrinsic noisy-in-origin source. By increasing the importance of the noise with respect to the herding, the system exhibits a finite-size phase transition from a quasiconsensus state, where most of the voters share the same opinion, to one with coexistence. Upon introducing some zealots, or voters with fixed opinion, the latter scenario may change significantly. We unveil new situations by carrying out a systematic numerical and analytical study of a fully connected network for voters, but allowing different voters to be directly influenced by different zealots. We show that this general system is equivalent to a system of voters without zealots, but with heterogeneous values of their parameters characterizing herding and noisy dynamics. We find excellent agreement between our analytical and numerical results. Noise and herding or zealotry acting together in the voter model yields a nontrivial mixture of the scenarios with the two mechanisms acting alone: It represents a situation where the global-local (noise-herding) competition is coupled to a symmetry breaking (zealots). In general, the zealotry enhances the effective noise of the system, which may destroy the original quasiconsensus state, and can introduce a bias towards the opinion of the majority of zealots, hence breaking the symmetry of the system and giving rise to new phases. In the most general case we find two different transitions: a discontinuous transition from an asymmetric bimodal phase to an extreme asymmetric phase and a second continuous transition from the extreme asymmetric phase to an asymmetric unimodal phase.

Critical noise can make the minority candidate win: The U.S. presidential election cases

A national voting population, when segmented into groups such as, for example, different states, can yield a counterintuitive scenario in which the winner may not necessarily get the highest number of total votes. A recent example is the 2016 presidential election in the United States. We model the situation by using interacting opinion dynamics models, and we look at the effect of coarse graining near the critical points where the spatial fluctuations are high. We establish that the sole effect of coarse graining, which mimics the "winner take all" electoral college system in the United States, can give rise to finite probabilities of events in which a minority candidate wins even in the large size limit near the critical point. The overall probabilities of victory of the minority candidate can be predicted from the models, which indicate that one may expect more instances of minority candidate winning in the future.

From synchronous to one-time delayed dynamics in coupled maps

We study the completely synchronized states (CSSs) of a system of coupled logistic maps as a function of three parameters: interaction strength (ɛ), range of the interaction (α), that can vary from first neighbors to global coupling, and a parameter (β) that allows one to scan continuously from nondelayed to one-time delayed dynamics. In the α-ɛ plane we identify periodic orbits, limit cycles, and chaotic trajectories, and describe how these structures change with delay. These features can be explained by studying the bifurcation diagrams of a two-dimensional nondelayed map. This allows us to understand the effects of one-time delays on CSSs, e.g., regularization of chaotic orbits and synchronization of short-range coupled maps, observed when the dynamics is moderately delayed. Finally, we substitute the logistic map with cubic and logarithmic maps, in order to test the robustness of our findings.

Symmetry breaking by heating in a continuous opinion model

We study the critical behavior of a continuous opinion model, driven by kinetic exchanges in a fully connected population. Opinions range in the real interval [-1,1], representing the different shades of opinions against and for an issue under debate. Individuals' opinions evolve through pairwise interactions, with couplings that are typically positive, but a fraction p of negative ones is allowed. Moreover, a social temperature parameter T controls the tendency of the individual responses toward neutrality. Depending on p and T, different collective states emerge: symmetry broken (one side wins), symmetric (tie of opposite sides), and absorbing neutral (indecision wins). We find the critical points and exponents that characterize the phase transitions between them. The symmetry breaking transition belongs to the usual Ising mean-field universality class, but the absorbing-phase transitions, with β=0.5, are out of the paradigmatic directed percolation class. Moreover, ordered phases can emerge by increasing social temperature.

Stochastic bifurcations in the nonlinear parallel Ising model

We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to that seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a "bubbling" behavior, as also happens for dilution.

Disorder-induced phase transition in an opinion dynamics model: Results in two and three dimensions

We study a model of continuous opinion dynamics with both positive and negative mutual interactions. 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 two and three 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 respective dimensions.

Complete synchronization equivalence in asynchronous and delayed coupled maps

Coupled map lattices are paradigmatic models of many collective phenomena. However, quite different patterns can emerge depending on the updating scheme. While in early versions, maps were updated synchronously, there has been in recent years a concern to consider more realistic updating schemes where elements do not change all at once. Asynchronous updating schemes and the inclusion of time delays are seen as more realistic than the traditional parallel dynamics, and, in diverse works, either one or the other has been implemented separately. But are they actually distinct cases? For coupled map lattices with adjustable range of interactions, we prove, using both numerical and analytical tools, that an adequate delayed dynamics leads to the same completely synchronized states as an asynchronous update, providing a unified framework to identify the stability conditions for complete synchronization.

Modeling Radicalization Phenomena in Heterogeneous Populations

The phenomenon of radicalization is investigated within a mixed population composed of core and sensitive subpopulations. The latest includes first to third generation immigrants. Respective ways of life may be partially incompatible. In case of a conflict core agents behave as inflexible about the issue. In contrast, sensitive agents can decide either to live peacefully adjusting their way of life to the core one, or to oppose it with eventually joining violent activities. The interplay dynamics between peaceful and opponent sensitive agents is driven by pairwise interactions. These interactions occur both within the sensitive population and by mixing with core agents. The update process is monitored using a Lotka-Volterra-like Ordinary Differential Equation. Given an initial tiny minority of opponents that coexist with both inflexible and peaceful agents, we investigate implications on the emergence of radicalization. Opponents try to turn peaceful agents to opponents driving radicalization. However, inflexible core agents may step in to bring back opponents to a peaceful choice thus weakening the phenomenon. The required minimum individual core involvement to actually curb radicalization is calculated. It is found to be a function of both the majority or minority status of the sensitive subpopulation with respect to the core subpopulation and the degree of activeness of opponents. The results highlight the instrumental role core agents can have to hinder radicalization within the sensitive subpopulation. Some hints are outlined to favor novel public policies towards social integration.

Inflexibility and independence: Phase transitions in the majority-rule model

In this work we study opinion formation in a population participating in a public debate with two distinct choices. We consider three distinct mechanisms of social interactions and individuals' behavior: conformity, nonconformity, and inflexibility. The conformity is ruled by the majority-rule dynamics, whereas the nonconformity is introduced in the population as an independent behavior, implying the failure of attempted group influence. Finally, the inflexible agents are introduced in the population with a given density. These individuals present a singular behavior, in a way that their stubbornness makes them reluctant to change their opinions. We consider these effects separately and all together, with the aim to analyze the critical behavior of the system. We perform numerical simulations in some lattice structures and for distinct population sizes. Our results suggest that the different formulations of the model undergo order-disorder phase transitions in the same universality class as the Ising model. Some of our results are complemented by analytical calculations.

Bifurcations in models of a society of reasonable contrarians and conformists

We study models of a society composed of a mixture of conformist and reasonable contrarian agents that at any instant hold one of two opinions. Conformists tend to agree with the average opinion of their neighbors and reasonable contrarians tend to disagree, but revert to a conformist behavior in the presence of an overwhelming majority, in line with psychological experiments. The model is studied in the mean-field approximation and on small-world and scale-free networks. In the mean-field approximation, a large fraction of conformists triggers a polarization of the opinions, a pitchfork bifurcation, while a majority of reasonable contrarians leads to coherent oscillations, with an alternation of period-doubling and pitchfork bifurcations up to chaos. Similar scenarios are obtained by changing the fraction of long-range rewiring and the parameter of scale-free networks related to the average connectivity.