The Effect of Exogenous Inputs and Defiant Agents on Opinion Dynamics With Local and Global Interactions

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.