The Annals of Applied Probability

Majority dynamics on trees and the dynamic cavity method

Yashodhan Kanoria and Andrea Montanari

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 5 (2011), 1694-1748.

Dates
First available in Project Euclid: 25 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1319576607

Digital Object Identifier
doi:10.1214/10-AAP729

Mathematical Reviews number (MathSciNet)
MR2884049

Zentralblatt MATH identifier
1266.60159

Subjects
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

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

Citation

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


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