The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 21, Number 5 (2011), 1694-1748.
Majority dynamics on trees and the dynamic cavity method
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.
Ann. Appl. Probab., Volume 21, Number 5 (2011), 1694-1748.
First available in Project Euclid: 25 October 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]
Secondary: 05C05: Trees 91A12: Cooperative games 91A26: Rationality, learning 91D99: None of the above, but in this section 93A14: Decentralized systems
Kanoria, Yashodhan; Montanari, Andrea. Majority dynamics on trees and the dynamic cavity method. Ann. Appl. Probab. 21 (2011), no. 5, 1694--1748. doi:10.1214/10-AAP729. https://projecteuclid.org/euclid.aoap/1319576607