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 $-1$. The time to escape this polarized state and reach consensus scales exponentially with population size.
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 consensus formation in interacting systems that evolve by multi-state majority rule and by plurality rule. In an update event, a group of G agents (with G odd), each endowed with an s-state spin variable, is specified. For majority rule, all group members adopt the local majority state; for plurality rule the group adopts the local plurality state. This update is repeated until a final consensus state is generally reached. In the mean field limit, the consensus time for an N-spin system increases as ln N for both majority and plurality rule, with an amplitude that depends on s and G. …
We define a stochastic reaction-diffusion process that describes a consensus formation in a non-sedentary population. The process is a diffusive version of the Majority Vote model, where the state update follows two stages: in the first stage, spins are allowed to hop to neighbor nodes with different probabilities for the respective spin orientation, and in the second stage, the spins in the same node can change its values according to the majority vote update rule. The model presents a consensus formation phase when concentration is greater than a threshold value, and a paramagnetic phase on …
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…
The statistical properties of pairwise majority voting over S alternatives is analyzed in an infinite random population. We first compute the probability that the majority is transitive (i.e. that if it prefers A to B to C, then it prefers A to C) and then study the case of an interacting population. This is described by a constrained multi-component random field Ising model whose ferromagnetic phase describes the emergence of a strong transitive majority. We derive the phase diagram, which is characterized by a tri-critical point and show that, contrary to intuition, it may be more likely for…
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…
Monte Carlo simulations of a Sznajd model show that if a near-consensus is formed out of four initially equally widespread opinions, the one which at intermediate times is second in the number of adherents usually loses out against the third-placed opinion.
We introduce a 2-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly-selected agents, consensus is reached in a time that scales ln N, where N is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of N. The upper critical dimension appears to be larger than 4. The final opinion a…
Street demonstrations occur across the world. In Rio de Janeiro, June/July 2013, they reach beyond one million people. A wrathful reader of \textit{O Globo}, leading newspaper in the same city, published a letter \cite{OGlobo} where many social questions are stated and answered Yes or No. These million people of street demonstrations share opinion consensus about a similar set of social issues. But they did not reach this consensus within such a huge numbered meetings. Earlier, they have met in diverse small groups where some of them could be convinced to change mind by other few fellows. Sudd…
In the Sznajd model of 2000, a pair of neighbouring agents on a square lattice convinces its six neighbours of the pair opinion iff the two agents of the pair share the same opinion. Now we replace the usual random sequential updating rule by simultaneous updating and find that this change makes a complete consenus much more difficult. The larger the lattice is, the higher must be the initial majority for one of the two competing opinions to become the consensus.
The dynamics of the model of agents with limited confidence introduced by Hegselmann and Krause exhibits multiple well-separated regimes characterised by the number of distinct clusters in the stationary state. We present indications that there are genuine dynamical phase transitions between these regimes. The main indicator is the divergence of the average evolution time required to reach the stationary state. The slowdown close to the transition is connected with the emergence of the groups of mediator agents which are very small but have decisive role in the process of social convergence. M…
We introduce a socially motivated extension of the voter model in which individual voters are also influenced by two opposing, fixed-opinion news sources. These sources forestall consensus and instead drive the population to a politically polarized state, with roughly half the population in each opinion state. Two types social networks for the voters are studied: (a) the complete graph of $N$ voters and, more realistically, (b) the two-clique graph with $N$ voters in each clique. For the complete graph, many dynamical properties are soluble within an annealed-link approximation, in which a lin…
We show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value p_c \approx 10%, there is a dramatic decrease in the time, T_c, taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < p_c, T_c \sim \exp(α(p)N), while for p > p_c, T_c \sim \ln N. We conclude with simulation results for Er…
A voter sits on each vertex of an infinite tree of degree $k$, and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large $k$-limit. This approach is inspired by the theory of mean field spin-g…
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 …
