Stochastic Opinion Dynamics Under Social Pressure in Arbitrary Networks

Social pressure is a key factor affecting the evolution of opinions on networks in many types of settings, pushing people to conform to their neighbors' opinions. To study this, the interacting Pólya urn model was introduced by Jadbabaie et al. [1], in which each agent has two kinds of opinion: inherent beliefs, which are hidden from the other agents and fixed; and declared opinions, which are randomly sampled at each step from a distribution which depends on the agent's inherent belief and her neighbors' past declared opinions (the social pressure component), and which is then communicated to their neighbors. Each agent also has a bias parameter denoting her level of resistance to social pressure. At every step, each agent updates her declared opinion (simultaneously with all other agents) according to her neighbors' aggregate past declared opinions, her inherent belief, and her bias parameter. We study the asymptotic behavior of this opinion dynamics model and show that agents' declaration probabilities converge almost surely in the limit using Lyapunov theory and stochastic approximation techniques. We also derive a sufficient condition for the agents to approach consensus on their declared opinions.