Phase transitions on fixed connected graphs and random graphs in the presence of noise

In this paper, we study phase transition behavior emerging from the interactions between 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 connected agents' values. 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; whereas at a noise level lower than the threshold, the system exhibits spontaneous symmetry breaking. We also verify the phase transition using simulations. This result may be found useful in the study of the collective behavior of complex systems under communication constraints.