  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 Polya urn model was
introduced by Jadbabaie et al., 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, the agents simultaneously update
their declared opinions according to their neighbors' aggregate past declared
opinions, their inherent beliefs, and their bias parameters. 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 necessary and
sufficient conditions for the agents to approach consensus on their declared
opinions. Our work provides further insight into the difficulty of inferring
the inherent beliefs of agents when they are under social pressure.
