  In this paper, we study the phase transition behavior emerging from the
interactions among multiple agents in the presence of noise. We propose a
simple discrete-time model in which a group of non-mobile agents form either a
fixed connected graph or a random graph process, and each agent, taking bipolar
value either +1 or -1, updates its value according to its previous value and
the noisy measurements of the values of the agents connected to it. We present
proofs for the occurrence of the following phase transition behavior: At a
noise level higher than some threshold, the system generates symmetric behavior
(vapor or melt of magnetization) or disagreement; whereas at a noise level
lower than the threshold, the system exhibits spontaneous symmetry breaking
(solid or magnetization) or consensus. The threshold is found analytically. The
phase transition occurs for any dimension. Finally, we demonstrate the phase
transition behavior and all analytic results using simulations. This result may
be found useful in the study of the collective behavior of complex systems
under communication constraints.
