Agreement in Spins and Social Networks

We introduce a model of mutually influencing individuals who hold one of two opinions about some matter. This is similar to a system of particles that can have one of two spin states. The model considers N sub-systems where individuals stay for some time, and then move from one subnetwork to another independently of each other, or they may leave the overall network from any one of the sub-systems. They arrive externally to any sub-system according to independent Poisson processes. In each sub-system individuals can influence each other to align with their own opinion or spin, and their opinion can also fluctuate at random in either opinion. The system also allows for a bias or directional field in any of the sub-systems that influences the individuals or spins that are locally present. We show that even with a weak bias, when random fluctuations become small then all the individuals or spins in a given sub-network will align with probability one, to the opinion or spin direction represented by the bias or spin.