Mathematics – Optimization and Control
Scientific paper
2008-08-24
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 8, 1817-1825, SEPTEMBER 2008
Mathematics
Optimization and Control
15 pages, 3 figures. To appear in the IEEE Transactions on Automatic Control
Scientific paper
10.1109/TAC.2008.929382
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.
Elia Nicola
Liu Haifeng
Liu Jialing
Olson Joshua M.
Sehgal Hullas
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