Phase Transitions on Fixed Connected Graphs and Random Graphs in the Presence of Noise

TL;DR

This paper analyzes how noise influences phase transitions in networks of agents, showing that below a certain noise threshold, the system reaches consensus or magnetization, while above it, disagreement or melting occurs, with results supported by analytical proofs and simulations.

Contribution

It introduces a simple discrete-time model for agent interactions on fixed or random graphs, providing analytical thresholds for phase transitions in noisy environments.

Findings

Threshold noise level for phase transition is analytically derived.

System exhibits symmetry breaking or consensus below the threshold.

Simulations confirm theoretical predictions across dimensions.

Abstract

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.