On majority voting games in trees

Abstract

We consider the following synchronous colouring game played on a simple connected graph with vertices coloured black or white. During one step of the game, each vertex is recoloured accordingto the majority of its neighbours. The variants of the model differ by the choice of a particular tie-breaking rule and possible rule for enforcing monotonicity. Two tie-breaking rules we consider are simple majority and strong majority, the first in case of a tie recolours the vertex black and the latter does not change the colour. The monotonicity-enforcing rule allows the voting only in white vertices, thus leavin gall black vertices intact. This model is called irreversible. These synchronous dynamic systems have been extensively studied and have many applications in molecular biology, distributed systems modelling, etc. In this paper we give two results describing the behaviour of these systems on trees. First we count the number of fix points of strong majority rule on complete binary trees to be asymptotically 4N ・ (2α)N where N is the number of vertices and 0.7685 ≤ α ≤ 0.7686. The second result is an algorithm for testing whether a given configuration on an arbitrary tree evolves into an all-black state under irreversible simple majority rule. The algorithm works in time O(t log t) where t is the number of black vertices and uses labels of length O(logN).