Best-of-Three Voting on Dense Graphs

03/22/2019

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by   Nan Kang, et al.

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Given a graph G of n vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Best-of-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree d = n^α , where α = Ω( ( n)^-1). We prove that if initially each vertex is red with probability greater than 1/2+δ, and blue otherwise, where δ≥ ( d)^-C for some C>0, then with high probability this dynamic reaches a final state where all vertices are red within O( n) + O( ( δ^-1) ) steps.