Majority dynamics on trees and the dynamic cavity method

An elector sits on each vertex of an infinite tree of degree k, and has to decide between two alternatives. At each time step, each elector 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 klimit. 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 non-trivial for small, odd values of k. 1 Definitions and main results 1.1 The majority process Consider a graph G with vertex set V, and edge set E . In the following, we shall denote by ∂i the set of neighbors of i ∈ V, and assume |∂i| < ∞ (i.e. G is locally finite). To each vertex i ∈ V we assign an initial spin σi(0) ∈ {−1,+1}. The vector of all initial spins is denoted by σ(0). Configuration σ(t) = {σi(t) : i ∈ V} at subsequent times t = 1, 2, . . . are determined according to the following majority update rule. If ∂i is the set of neighbors of node i ∈ V, we let σi(t+ 1) = sign (∑ j∈∂i σj(t) ) (1) when ∑ j∈∂i σj(t) 6= 0. If ∑ j∈∂i σj(t) = 0, then we let σi(t+ 1) = { σi(t) with probability 1/2, −σi(t) with probability 1/2. (2) In order to construct this process, we associate to each vertex i ∈ V, a sequence of i.i.d. Bernoulli(1/2) random variables Ai = {Ai,0, Ai,1, Ai,2 . . . }, whereby Ai,t is used to break the (eventual) tie at time t. A realization of the process is then determined by the triple (G,A, σ(0)), with A = {Ai}. In this work we will study the asymptotic dynamic of this process when G is an infinite regular tree of degree k ≥ 2. Let Pθ be the law of the majority process where, in the initial configuration, the spins σi(0) Department of Electrical Engineering, Stanford University Department of Electrical Engineering and Department of Statistics, Stanford University