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

Jialing Liu, Vikas Yadav, Hullas Sehgal, Joshua M. Olson, Haifeng Liu, Nicola Elia

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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 -, 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.

Original languageEnglish (US)
Pages (from-to)1817-1825
Number of pages9
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - 2008
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received January 12, 2006; revised November 29, 2007. First published September 12, 2008; current version published September 24, 2008. This paper was presented in part at the 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC’05). This work was supported in part by the National Science Foundation under Grant ECS-0093950 and by the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA. Recommended by Associate Editor C. Abdallah.


  • Consensus
  • Limited communication
  • Networked dynamical systems
  • Phase transitions
  • Random graphs


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