The Annals of Applied Probability

Majority dynamics on trees and the dynamic cavity method

Yashodhan Kanoria and Andrea Montanari

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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

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

First available in Project Euclid: 25 October 2011

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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

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


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.

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