Majority-rule Dynamics Research Articles

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 T<Tc, where the population's mean opinion m oscillates around a positive or a negative value from a unimodal oscillatory regime for T>Tc 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.

Divergence and consensus in majority rule

We investigate majority rule dynamics in a population with two classes of people, each with two opinion states ±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 rate ε. Consensus is achieved in a time that scales logarithmically with population size if ε≥ε_{c}=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 -1. The time to escape this polarized state and reach consensus scales exponentially with population size.

Organisational Social Influence on Directed Hierarchical Graphs, from Tyranny to Anarchy

Coordinated human behaviour takes place within a diverse range of social organisational structures, which can be thought of as power structures with “managers” who influence “subordinates”. A change in policy in one part of the organisation can cause cascades throughout the structure, which may or may not be desirable. As organisations change in size, complexity and structure, the system dynamics also change. Here, we consider majority rule dynamics on organisations modelled as hierarchical directed graphs, where the directed edges indicate influence. We utilise a topological measure called the trophic incoherence parameter, q, which effectively gauges the stratification of power structure in an organisation. We show that this measure bounds regimes of behaviour. There is fast consensus at low q (e.g. tyranny), slow consensus at mid q (e.g. democracy), and no consensus at high q (e.g. anarchy). These regimes are investigated analytically, numerically and empirically with diverse case studies in the Roman Army, US Government, and a healthcare organisation. Our work demonstrates the usefulness of the trophic incoherence parameter when considering models of social influence dynamics, with widespread consequences in the design and analysis of organisations.

Majority rule dynamics between a double coalition and a third opinion: coalition profit models and majority coalition ties

This article explores the opinion dynamics of a double coalition opinion against a third opinion under majority rule updates on odd fixed size connected groups. For this purpose, coalition benefit criteria and three opinion formation models which extend the 2-state majority rule model on lattices are introduced. The proposed models focus on the coalition profit of its constituent coalition opinions and cover the possible final scenarios from coalition alliance perspective: either minor opinion or major opinion is favored, or dynamics do not favor to any coalition opinion. Opinion exchanges take place on a torus embedded lattice network of a 3-state system having in consideration tie configurations and two rules to break them: either by random choice or leaving ties unaltered. Models were analyzed in the statistical mechanics spirit through Monte Carlo simulations without node replacement. Estimations for coalition benefits, the growth of coalition ties, and consensus probabilities are reported. The loss of coalition strengths due to coalition ties and its indecision is indicated. In particular, the logistic decay of consensus probability is due to the logistic adaptive growth of coalition ties. Scaling behaviors for consensus time and coalition ties in terms of network size are suggested. The results of numerical simulations are discussed in the context of social influence and social dynamics.

Dynamics of epidemic spreading with vaccination: Impact of social pressure and engagement

In this work we consider a model of epidemic spreading coupled with an opinion dynamics in a fully-connected population. Regarding the opinion dynamics, the individuals may be in two distinct states, namely in favor or against a vaccination campaign. Individuals against the vaccination follow a standard SIS model, whereas the pro-vaccine individuals can also be in a third compartment, namely Vaccinated. In addition, the opinions change according to the majority-rule dynamics in groups with three individuals. We also consider that the vaccine can give permanent or temporary immunization to the individuals. By means of analytical calculations and computer simulations, we show that the opinion dynamics can drastically affect the disease propagation, and that the engagement of the pro-vaccine individuals can be crucial for stopping the epidemic spreading. The full numerical code for simulate the model is available from the authors' webpage.

Majority Rule Based Opinion Dynamics with Biased and Stubborn Agents

In this paper, we investigate the impact of majority-rule based random interactions among agents in a large social network on the diffusion of opinions in the network. Opinion of each agent is assumed to be a binary variable taking values in the set {0, 1}. Interactions among agents are modeled using the majority rule, where each agent updates its opinion at random instants by adopting the ' majority ' opinion among a group of randomly sampled agents. We investigate two scenarios that respectively incorporate `bias' of the agents towards a specific opinion and stubbornness of some of the agents in the majority rule dynamics. For the first scenario, where all the agents are assumed to be ' biased ' towards one of the opinions, it is shown that the agents reach a consensus on the preferred opinion (with high probability) only if the initial fraction of agents having the preferred opinion is above a certain threshold. Furthermore, the mean time taken to reach the consensus is shown to be logarithmic in the network size. In the second scenario, where the presence of ' stubborn ' agents, who never update their opinions, is assumed, we characterize the equilibrium distribution of opinions of the non-stubborn agents using mean field techniques. The mean field limit is shown to have multiple stable equilibrium points which leads to a phenomenon known as metastability .

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.

Effect of Heterogeneity in Initial Geographic Distribution on Opinions’ Competitiveness

Spin dynamics on networks allows us to understand how a global consensus emerges out of individual opinions. Here, we are interested in the effect of heterogeneity in the initial geographic distribution of a competing opinion on the competitiveness of its own opinion. Accordingly, in this work, we studied the effect of spatial heterogeneity on the majority rule dynamics using a three-state spin model, in which one state is neutral. Monte Carlo simulations were performed on square lattices divided into square blocks (cells). Accordingly, one competing opinion was distributed uniformly among cells, whereas the spatial distribution of the rival opinion was varied from the uniform to heterogeneous, with the median-to-mean ratio in the range from 1 to 0. When the size of discussion group is odd, the uncommitted agents disappear completely after 3.30 ± 0.05 update cycles, and then the system evolves in a two-state regime with complementary spatial distributions of two competing opinions. Even so, the initial heterogeneity in the spatial distribution of one of the competing opinions causes a decrease of this opinion competitiveness. That is, the opinion with initially heterogeneous spatial distribution has less probability to win, than the opinion with the initially uniform spatial distribution, even when the initial concentrations of both opinions are equal. We found that although the time to consensus , the opinion’s recession rate is determined during the first 3.3 update cycles. On the other hand, we found that the initial heterogeneity of the opinion spatial distribution assists the formation of quasi-stable regions, in which this opinion is dominant. The results of Monte Carlo simulations are discussed with regard to the electoral competition of political parties.

Answer to the Commentary on "A note on the consensus time of mean-field majority-rule dynamics"

A Reply to the Commentary by Hugo Fort [Pap. Phys. 1, 010003 (2009)].Received: 7 October 2009, Accepted: 7 October 2009; Edited by: M. C. Barbosa; DOI: 10.4279/PIP.010004

Commentary on "A note on the consensus time of mean-field majority-rule dynamics"

A Commentary on the paper by D. H. Zanette [Pap. Phys. 1, 010002 (2009)]. The author of the paper offers a Reply.Received: 2 October 2009, Accepted: 6 October 2009; Edited by: M. C. Barbosa; DOI: 10.4279/PIP.010003

A note on the consensus time of mean-field majority-rule dynamics

In this work, it is pointed out that in the mean-field version of majority-rule opinion dynamics the dependence of the consensus time on the population size exhibits two regimes. This is determined by the size distribution of the groups that, at each evolution step, gather to reach agreement. When the group size distribution has a finite mean value, the previously known logarithmic dependence on the population size holds. On the other hand, when the mean group size diverges, the consensus time and the population size are related through a power law. Numerical simulations validate this semi-quantitative analytical prediction.Received: 6 July 2009, Accepted: 2 September 2009; Edited by: M. C. Barbosa; Reviewed by: H. Fort, Universidad de la República, Uruguay; DOI: 10.4279/PIP.010002

Activity patterns on random scale-free networks: global dynamics arising from localmajority rules

Activity or spin patterns on a random scale-free network are studied using meanfield analysis and computer simulations. These activity patterns evolve in timeaccording to local majority rule dynamics which is implemented using (i)parallel or synchronous updating and (ii) random sequential or asynchronousupdating. Our mean field calculations predict that the relaxation processes ofdisordered activity patterns become much more efficient as the scaling exponentγ of the scale-free degree distribution changes fromγ > 5/2 toγ < 5/2.For γ > 5/2, the corresponding decay times increase asln(N) with increasingnetwork size N whereasthey are independent of N for γ < 5/2. In order to check these mean field predictions, extensive simulations of the patterndynamics have been performed using two different ensembles of random scale-freenetworks: (A) multi-networks as generated by the configuration method, whichtypically leads to many self-connections and multiple edges, and (B) simplenetworks without self-connections and multiple edges. We find that the mean fieldpredictions are confirmed (i) for random sequential updating of multi-networks and(ii) for both parallel and random sequential updating of simple networks withγ = 2.25 and2.6. For γ = 2.4, the data for the simple networks seem to be consistent with mean field theory as well,whereas we cannot draw a definite conclusion from the simulation data for the multi-networks.The latter difficulty can be understood in terms of an effective scaling exponentγeff = γeff(γ, N) for multi-networks. This effective exponent is determined byremoving all self-connections and multiple edges; it satisfiesγeff ≥ γ and decreases towardsγ with increasingnetwork size N. For γ = 2.4, we find γeff ≳ 5/2 up to N = 217.

Majority rule dynamics in finite dimensions

We investigate the long-time behavior of a majority rule opinion dynamics model in finite spatial dimensions. Each site of the system is endowed with a two-state spin variable that evolves by majority rule. In a single update event, a group of spins with a fixed (odd) size is specified and all members of the group adopt the local majority state. Repeated application of this update step leads to a coarsening mosaic of spin domains and ultimate consensus in a finite system. The approach to consensus is governed by two disparate time scales, with the longer time scale arising from realizations in which spins organize into coherent single-opinion bands. The consequences of this geometrical organization on the long-time kinetics are explored.

Local Simple Games in Public Choice Mechanisms

This paperinitiates an investigation of processes in wlhiclh cooperative games determine at each moment the directioni of motion of the social state. Attentioln is focused upoIn simple games, i.e., uponl games in which a coalitioni cani prevenit either every direction orno direction from beinig choseni. As motivation, conisider the problem of COInStlruCtinlg a planning pr^ocedule to allocate goods. Existing procedures, as surveyed in Tulkenis [1978], specify a local (instantaneous) game that determnines the direction of change of the allocation vector. For example, the MDP procedure of Malinvaud [1970-1] and Dreze and de la Valee Poussin [1971] specifies the direction of change at each moment as a function of reported marginal rates of substitution. Assuming that individuals are concerned only with mnaximizing thleir rates of utility increase, local games can be constructed so that their diiectionial equilibria will lead the allocation vector to the Pareto set. Various equiilibriuml concepts have been used. In MDP local games, for exanmple, maxmiin equilibria (Dreze and de la Valee Poussin [1971]), Nashi equilibria (Roberts [1979], SchoLumaker [1979]), and certain cooperative equilibria (Tulkens and Zamir [1979]) have been shown to cause convergence to the Pareto set. Whenever a procedure is actually to be implemented, account should be taken of the fact that it will be constrained by rules that are effective in the society. For example, in a dictatorship the social state must move in a direction that is most preferred by the dictator; the dictator will simply not abide by the equilibrium direction of a local game unless it is one of his most preferred directions. Less trivially, whenever unanimous disapproval of a motion can prevent its occurrence, the society will only agree to a procedure whose equilibrium direction of motion cannot be feasibly changed to a direction that increases everybody's utility at a greater rate. These two examples of constrainits on motion are not of the type that is usually imposed in the literature on allocation procedures and, furthermore, in areas like tax reform (e.g., Guesnerie [1977]) and majority rule dynamics (e.g., McKelvey [1976], Schofield [1978a]). The constraints oni motion in these studies call be viewed as resulting from a requirement that any motioni must increase the utility