Most of the conventional models for opinion dynamics mainly account for a fully local influence, where myopic agents decide their actions after they interact with other agents that are adjacent to them. For example, in the case of social interactions, this includes family, friends, and other immediate strong social ties. The model proposed in this paper embodies a global influence as well where by global we mean that each node also observes a sample of the average behavior of the entire population; e.g., in the social example, people observe other people on the streets, subway, and other social venues. We consider the case where nodes have dichotomous states; examples of applications include elections with two major parties, whether or not to adopt a new technology or product, and any yes no opinion such as in voting on a referendum. The dynamics of states on a network with arbitrary degree distribution are studied. For a given initial condition, we find the probability to reach consensus on each state and the expected time reach to consensus. To model mass media, the effect of an exogenous bias on the average orientation of the system is investigated. To do so, we add an external field to the model that favors one of the states over the other. This field interferes with the regular decision process of each node and creates a constant probability to lean towards one of the states. We solve for the average state of the system as a function of time for given initial conditions. Then anti-conformists (stubborn nodes who never revise their states) are added to the network, in an effort to circumvent the external bias. We find necessary conditions on the number of these defiant nodes required to cancel the effect of the external bias. Our analysis is based on a mean field approximation of the agent opinions.
 A simple Ising spin model which can describe a mechanism of making a decision in a closed community is proposed. It is shown via standard Monte Carlo simulations that very simple rules lead to rather complicated dynamics and to a power law in the decision time distribution. It is found that a closed community has to evolve either to a dictatorship or a stalemate state (inability to take any common decision). A common decision can be taken in a "democratic way" only by an open community.  In 2000 we proposed a sociophysics model of opinion formation, which was based on trade union maxim "United we Stand, Divided we Fall" (USDF) and latter due to Dietrich Stauffer became known as the Sznajd model (SM). The main difference between SM compared to voter or Ising-type models is that information flows outward. In this paper we review the modifications and applications of SM that have been proposed in the literature. We introduce a two-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 @math , where @math 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 @math . The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension.   We introduce and study the reverse voter model, a dynamics for spin variables similar to the well‐known voter dynamics. The difference is in the way neighbors influence each other: once a node is selected and one among its neighbors chosen, the neighbor is made equal to the selected node, while in the usual voter dynamics the update goes in the opposite direction. The reverse voter dynamics is studied analytically, showing that on networks with degree distribution decaying as k−v, the time to reach consensus is linear in the system size N for all v > 2. The consensus time for link‐update voter dynamics is computed as well. We verify the results numerically on a class of uncorrelated scale‐free graphs.   In this work we consider the influence of mass media in the dynamics of the two-dimensional Sznajd model. This influence acts as an external field, and it is introduced in the model by means of a probability p of the agents to follow the media opinion. We performed Monte Carlo simulations on square lattices with different sizes, and our numerical results suggest a change on the critical behavior of the model, with the absence of the usual phase transition for p>∼0.18. Another effect of the probability p is to decrease the average relaxation times τ, that are log-normally distributed, as in the standard model. In addition, the τ values depend on the lattice size L in a power-law form, τ∼Lα, where the power-law exponent depends on the probability p.