Open Access
October 2011 Majority dynamics on trees and the dynamic cavity method
Yashodhan Kanoria, Andrea Montanari
Ann. Appl. Probab. 21(5): 1694-1748 (October 2011). DOI: 10.1214/10-AAP729


A voter sits on each vertex of an infinite tree of degree k, and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables.

In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large k-limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of k.


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Yashodhan Kanoria. Andrea Montanari. "Majority dynamics on trees and the dynamic cavity method." Ann. Appl. Probab. 21 (5) 1694 - 1748, October 2011.


Published: October 2011
First available in Project Euclid: 25 October 2011

zbMATH: 1266.60159
MathSciNet: MR2884049
Digital Object Identifier: 10.1214/10-AAP729

Primary: 60K35 , 82C22
Secondary: 05C05 , 91A12 , 91A26 , 91D99 , 93A14

Keywords: best response dynamics , consensus , dynamic cavity method , Ising spin dynamics , Majority dynamics , parallel/synchronous dynamics , social learning , trees

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.21 • No. 5 • October 2011
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