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.
We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of @math nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus @math scales as @math , where @math is the @math th moment of the degree distribution. For a power-law degree distribution @math , @math thus scales as @math for @math , as @math for @math , as @math for @math , as @math for @math , and as @math for @math . These results agree with simulation data for networks with both uncorrelated and correlated node degrees. We introduce the heterogeneous voter model (HVM), in which each agent has its own intrinsic rate to change state, reflective of the heterogeneity of real people, and the partisan voter model (PVM), in which each agent has an innate and fixed preference for one of two possible opinion states. For the HVM, the time until consensus is reached is much longer than in the classic voter model. For the PVM in the mean-field limit, a population evolves to a preference-based state, where each agent tends to be aligned with its internal preference. For finite populations, discrete fluctuations ultimately lead to consensus being reached in a time that scales exponentially with population size.