We study the effects of free will and massive opinion of multi-agents in a majority rule model wherein the competition of the two types of opinions is taken into account. To address this issue, we consider two specific models (model I and model II) involving different opinion-updating dynamics. During the opinion-updating process, the agents either interact with their neighbors under a majority rule with probability $1-q$, or make their own decisions with free will (model I) or according to the massive opinion (model II) with probability $q$. We investigate the difference of the average numbers of the two opinions as a function of $q$ in the steady state. We find that the location of the order-disorder phase transition point may be shifted according to the involved dynamics, giving rise to either smooth or harsh conditions to achieve an ordered state. For the practical case with a finite population size, we conclude that there always exists a threshold for $q$ below which a full consensus phase emerges. Our analytical estimations are in good agreement with simulation results.
Asymmetry in contrarian behavior is investigated within the Galam model of opinion dynamics using update groups of size 3 with two competing opinions A and B. Denoting $x$ and $y$ the respective proportions of A and B contrarians, four schemes of implementations are studied. First scheme activates contrarians after each series of updates with probabilities $x$ and $y$ for agents holding respectively opinion A and B. Second scheme activates contrarians within the update groups only against global majority with probability $x$ when A is majority and $y$ when B is majority. Third scheme considers…
We investigate the dynamics of the majority-rule opinion formation model when voters experience differential latencies. With this extension, voters that just adopted an opinion go into a latent state during which they are excluded from the opinion formation process. The duration of the latent state depends on the opinion adopted by the voter. The net result is a bias towards consensus on the opinion that is associated with the shorter latency. We determine the exit probability and time to consensus for systems of $N$ voters. Additionally, we derive an asymptotic characterisation of the time to…
We investigate the role of opinion leaders or influentials in the collective behavior of a social system. Opinion leaders are characterized by their unidirectional influence on other agents. We employ a model based on Axelrod's dynamics for cultural interaction among social agents that allows for non-interacting states. We find three collective phases in the space of parameters of the system, given by the fraction of opinion leaders and a quantity representing the number of available states: one ordered phase having the state imposed by the leaders; another nontrivial ordered phase consisting …
The complexity of human behaviour can lead to very unpredictable patterns in social activity and structure. Here we demonstrate the instability of a community network controlled by majority ruling, where an element adopts the most popular opinion of their peers. We modelled a community as a square lattice, and performed sequential time step numerical calculations upon each cell in parallel. Depending on the initial ratio of two opinions, the community can segregate either into separate gangs and cliques, or get dominated by a single opinion. We also note that gangs are separated by neutral or …
We study a nonequilibrium model with up-down symmetry and a noise parameter $q$ known as majority-vote model of M.J. Oliveira $1992$ on opinion-dependent network or Stauffer-Hohnisch-Pittnauer networks. By Monte Carlo simulations and finite-size scaling relations the critical exponents $β/ν$, $γ/ν$, and $1/ν$ and points $q_{c}$ and $U^*$ are obtained. After extensive simulations, we obtain $β/ν=0.230(3)$, $γ/ν=0.535(2)$, and $1/ν=0.475(8)$. The calculated values of the critical noise parameter and Binder cumulant are $q_{c}=0.166(3)$ and $U^*=0.288(3)$. Within the error bars, the exponents obe…
In this work we study a Sznajd-like opinion dynamics on a square lattice of linear size $L$. For this purpose, we consider that each agent has a convincing power $C$, that is a time-dependent quantity. Each high convincing power group of four agents sharing the same opinion may convince its neighbors to follow the group opinion, which induces an increase of the group's convincing power. In addition, we have considered that a group with a local majority opinion (3 up/1 down spins or 1 up/3 down spins) can persuade the agents neighboring the group with probability $p$, since the group's convinci…
We study how groups reach consensus by varying communication network structure and individual incentives. In 342 networks of seven individuals, single opinionated "leaders" can drive decision outcomes, but do not accelerate consensus formation, whereas conflicting opinions slow consensus. While networks with more links reach consensus faster, this advantage disappears under conflict. Unopinionated individuals make choices consistent with a local majority rule combined with "inertia" favouring their previous choice, while opinionated individuals favour their preferred option but yield under hig…
We revisit the no tie breaking 2-state Galam contrarian model of opinion dynamics for update groups of size 3. While the initial model assumes a constant density of contrarians a for both opinions, it now depends for each opinion on its global support. Proportionate contrarians are thus found to indeed preserve the former case main results. However, restricting the contrarian behavior to only the current collective majority, makes the dynamics more complex with novel features. For a density a<a_c=1/9 of one-sided contrarians, a chaotic basin is found in the fifty-fifty region separated from tw…
The existence of juxtaposed regions of distinct cultures in spite of the fact that people's beliefs have a tendency to become more similar to each other's as the individuals interact repeatedly is a puzzling phenomenon in the social sciences. Here we study an extreme version of the frequency-dependent bias model of social influence in which an individual adopts the opinion shared by the majority of the members of its extended neighborhood, which includes the individual itself. This is a variant of the majority-vote model in which the individual retains its opinion in case there is a tie among …
Majority-vote model is a much-studied model for social opinion dynamics of two competing opinions. With the recent appreciation that our social network comprises a variety of different layers forming a multiplex network, a natural question arises on how such multiplex interactions affect the social opinion dynamics and consensus formation. Here, the majority-vote model will be studied on multiplex networks to understand the effect of multiplexity on opinion dynamics. We will discuss how global consensus is reached by different types of voters: AND- and OR-rule voters on multiplex-network and v…
We study the effect of inflexible agents on two state opinion dynamics. The model operates via repeated local updates of random grouping of agents. While floater agents do eventually flip their opinion to follow the local majority, inflexible agents keep their opinion always unchanged. It is a quenched individual opinion. In the bare model (no inflexibles), a separator at 50% drives the dynamics towards either one of two pure attractors, each associated with a full polarization along one of the opinions. The initial majority wins. The existence of inflexibles for only one of the two opinions i…
The dynamics of opinion formation in a society is a complex phenomenon where many variables play an important role. Recently, the influence of algorithms to filter which content is fed to social networks users has come under scrutiny. Supposedly, the algorithms promote marketing strategies, but can also facilitate the formation of filters bubbles in which a user is most likely exposed to opinions that conform to their own. In the two-state majority-vote model an individual adopts an opinion contrary to the majority of its neighbors with probability $q$, defined as the noise parameter. Here, we…
We investigate a majority-vote model on two-layer multiplex networks with community structure. In our majority-vote model, the edges on each layer encode one type of social relationship and an individual changes their opinion based on the majority opinions of their neighbors in each layer. To capture the fact that different relationships often have different levels of importance, we introduce a layer-preference parameter, which determines the probability of a node to adopt an opinion when the node's neighborhoods on the two layers have different majority opinions. We construct our networks so …
Modelling efforts in opinion dynamics have to a large extent ignored that opinion exchange between individuals can also have an effect on how willing they are to express their opinion publicly. Here, we introduce a model of public opinion expression. Two groups of agents with different opinion on an issue interact with each other, changing the willingness to express their opinion according to whether they perceive themselves as part of the majority or minority opinion. We formulate the model as a multi-group majority game and investigate the Nash equilibria. We also provide a dynamical systems…
We study a model of a population making a binary decision based on information spreading within the population, which is fully connected or covering a square grid. We assume that a fraction of the population wants to make the choice of the majority, while the rest want to make the minority choice. This resembles opinion spreading with "contrarian" agents, but has the game theoretic aspect that agents try to optimize their own situation in ways that are incompatible with the common good. When this fraction is less than 1/2, the population can efficiently self-organize to a state where agents ge…
In this article, I investigate the use of Bayesian updating rules applied to modeling social agents in the case of continuos opinions models. Given another agent statement about the continuous value of a variable $x$, we will see that interesting dynamics emerge when an agent assigns a likelihood to that value that is a mixture of a Gaussian and a Uniform distribution. This represents the idea the other agent might have no idea about what he is talking about. The effect of updating only the first moments of the distribution will be studied. and we will see that this generates results similar t…
We investigate majority rule dynamics in a population with two classes of people, each with two opinion states $\pm 1$, and with tunable interactions between people in different classes. In an update, a randomly selected group adopts the majority opinion if all group members belong to the same class; if not, majority rule is applied with probability $ε$. Consensus is achieved in a time that scales logarithmically with population size if $ε\geq ε_c=\frac{1}{9}$. For $ε<ε_c$, the population can get trapped in a polarized state, with one class preferring the $+1$ state and the other preferring $-…
Traditional models of opinion dynamics provide a simple approach to understanding human behavior in basic social scenarios. However, when it comes to issues such as polarization and extremism, we require a more nuanced understanding of human biases and cognitive tendencies. In this paper, we propose an approach to modeling opinion dynamics by integrating mental models and assumptions of individuals agents using Bayesian-inspired methods. By exploring the relationship between human rationality and Bayesian theory, we demonstrate the efficacy of these methods in describing how opinions evolve. O…
We assume a community whose members adopt one of two opinions $A$ or $B$. Each member appears as an inflexible, or as a non-contrarian or contrarian floater. An inflexible sticks to its opinion, whereas a floater may change into a floater of the alternative opinion. The occurrence of this change is governed by the local majority rule: members meet in groups of a fixed size, and a floater then changes its opinion provided it is a minority in the group. Subsequently, a non-contrarian floater keeps the opinion as adopted under the local majority rule, whereas a contrarian floater adopts the alter…
We propose a model that extends the binary ``united we stand, divided we fall'' opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are …
A simple model of opinion formation dynamics in which binary-state agents make up their opinions due to the influence of agents in a local neighborhood is studied using different network topologies. Each agent uses two different strategies, the Sznajd rule with a probability $q$ and the Galam majority rule (without inertia) otherwise; being $q$ a parameter of the system. Initially, the binary-state agents may have opinions (at random) against or in favor about a certain topic. The time evolution of the system is studied using different network topologies, starting from different initial opinio…
We introduce a statistical physics model for opinion dynamics on random networks where agents adopt the opinion held by the majority of their direct neighbors only if the fraction of these neighbors exceeds a certain threshold, p_u. We find a transition from total final consensus to a mixed phase where opinions coexist amongst the agents. The relevant parameters are the relative sizes in the initial opinion distribution within the population and the connectivity of the underlying network. As the order parameter we define the asymptotic state of opinions. In the phase diagram we find regions of…
The Majority Rule is applied to a topology that consists of two coupled random networks, thereby mimicking the modular structure observed in social networks. We calculate analytically the asymptotic behaviour of the model and derive a phase diagram that depends on the frequency of random opinion flips and on the inter-connectivity between the two communities. It is shown that three regimes may take place: a disordered regime, where no collective phenomena takes place; a symmetric regime, where the nodes in both communities reach the same average opinion; an asymmetric regime, where the nodes i…
We explore the possibilities of enforcing and preventing consensus in continuous opinion dynamics that result from modifications in the communication rules. We refer to the model of Weisbuch and Deffuant, where $n$ agents adjust their continuous opinions as a result of random pairwise encounters whenever their opinions differ not more than a given bound of confidence $\eps$. A high $\eps$ leads to consensus, while a lower $\eps$ leads to a fragmentation into several opinion clusters. We drop the random encounter assumption and ask: How small may $\eps$ be such that consensus is still possible …
A two state model for opinion forming, which has proven heuristic power, is reviewed with a novel emphasis on the existence or absence of a threshold for the dynamics. Monitored by repeated small groups discussions floater agents update their opinion according to a local majority rule. A threshold makes the initial supports to flow towards eitheir one of two opposite attractors with only one single opinion. While odd sizes yield a threshold at fifty percent even sizes, which allow the inclusion of doubt at an opinion tie, produces a threshold shift toward either 0 or 1 giving rise to minority …
We study the effects of inhomogeneous influence of individuals on collective phenomena. We focus analytically on a typical model of the majority rule, applied to the completely connected agents. Two types of individuals $A$ and $B$ with different influence activity are introduced. The individuals $A$ and $B$ are distributed randomly with concentrations $ν$ and $1-ν$ at the beginning and fixed further on. Our main result is that the location of the order-disorder transition is affected due to the introduction of the inhomogeneous influence. This result highlights the importance of inhomogeneous…
A model where agents show discrete behavior regarding their actions, but have continuous opinions that are updated by interacting with other agents is presented. This new updating rule is applied to both the voter and Sznajd models for interaction between neighbors and its consequences are discussed. The appearance of extremists is naturally observed and it seems to be a characteristic of this model.
Non-linear voter models assume that the opinion of an agent depends on the opinions of its neighbors in a non-linear manner. This allows for voting rules different from majority voting. While the linear voter model is known to reach consensus, non-linear voter models can result in the coexistence of opposite opinions. Our aim is to derive approximations to correctly predict the time dependent dynamics, or at least the asymptotic outcome, of such local interactions. Emphasis is on a probabilistic approach to decompose the opinion distribution in a second-order neighborhood into lower-order prob…
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 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 bifurcatio…
In this work we study the majority-vote model with the presence of two distinc noises. The first one is the usual noise $q$, that represents the probability that a given agent follows the minority opinion of his/her social contacts. On the other hand, we consider the independent behavior, such that an agent can choose his/her own opinion $+1$ or $-1$ with equal probability, independent of the group's norm. We study the impact of the presence of such two kinds of stochastic driving in the phase transitions of the model, considering the mean field and the square lattice cases. Our results sugges…
In this work we study opinion formation in a population participating of a public debate with two distinct choices. We considered 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 to 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 stub…
Effect of higher-order interactions on noisy majority-rule dynamics with random group sizes
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Research Center for Quantum Physics, BRIN（BRIN量子物理研究中心）；Department of Physics, POSTECH（浦项工科大学物理系）；Department of Physics, Korea University（韩国大学物理系）；Department of Physics, Universitas Gadjah Mada（加查马达大学物理系）
We consider the process of opinion formation, in a society where there is a set of rules, $B$. These rules change over time due to the drift of public opinion, driven in part by publicity campaigns. Public opinion is formed by the integration of the voters' attitudes which can be either conservative (in agreement with $B$) or liberal (in agreement with peer voters). These attitudes are represented in the phase space of the system by stable fixed points. In the present letter we study the properties that an official publicity campaign must have in order to turn the public opinion in favor of $B…
The equilibrium states associated with the local majority-rule are divided into three classes, the states of system-wide coordination, the trapped states, and the states of period-2. The effect of clustering coefficient on the occurrence probability of the states of three classes is analyzed numerically for Watts-Strogatz and scale-free networks. We further study the effect of clustering coefficient on the robustness for the states of each class by proposing a stochastic local majority-rule. The states of period-2 are found to be easy to break up, and the trapped states are most robust among t…
We show how appropriate rewiring with the aid of Metropolis Monte Carlo computational experiments can be exploited to create network topologies possessing prescribed values of the average path length (APL) while keeping the same connectivity degree and clustering coefficient distributions. Using the proposed rewiring rules we illustrate how the emergent dynamics of the celebrated majority-rule model are shaped by the distinct impact of the APL attesting the need for developing efficient algorithms for tuning such network characteristics.
Dynamic social opinion models have been widely studied on undirected networks, and most of them are based on spin interaction models that produce a consensus. In reality, however, many networks such as Twitter and the World Wide Web are directed and are composed of both unidirectional and bidirectional links. Moreover, from choosing a coffee brand to deciding who to vote for in an election, two or more competing opinions often coexist. In response to this ubiquity of directed networks and the coexistence of two or more opinions in decision-making situations, we study a non-consensus opinion mo…
In this paper, we study how an agent's belief is affected by her neighbors in a social network. We first introduce a general framework, where every agent has an initial belief on a statement, and updates her belief according to her and her neighbors' current beliefs under some belief evolution functions, which, arguably, should satisfy some basic properties. Then, we focus on the majority rule belief evolution function, that is, an agent will (dis)believe the statement iff more than half of her neighbors (dis)believe it. We consider some fundamental issues about majority rule belief evolution,…
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…
We study the effects of social influences in opinion dynamics. In particular, we define a simple model, based on the majority rule voting, in order to consider the role of conformity. Conformity is a central issue in social psychology as it represents one of people's behaviors that emerges as a result of their interactions. The proposed model represents agents, arranged in a network and provided with an individual behavior, that change opinion in function of those of their neighbors. In particular, agents can behave as conformists or as nonconformists. In the former case, agents change opinion…
People are often divided into conformists and contrarians, the former tending to align to the majority opinion in their neighborhood and the latter tending to disagree with that majority. In practice, however, the contrarian tendency is rarely followed when there is an overwhelming majority with a given opinion, which denotes a social norm. Such reasonable contrarian behavior is often considered a mark of independent thought, and can be a useful strategy in financial markets. We present the opinion dynamics of a society of reasonable contrarian agents. The model is a cellular automaton of Isin…
