Size bounds for dynamic monopolies

Abstract

This paper considers a repetitive polling game played on an n-vertex graph G. Initially, each vertex is colored black or white. At each round, each vertex (simultaneously) recolors itself by the color of the majority of its neighborhood. A set of vertices M is said to be a dynamic monopoly, abbreviated dynamo, if starting the game with the vertices of M colored white, the system eventually reaches an all-white global state. We study the question of how small a dynamic monopoly might be in various models. In a number of these models we derive tight bounds of Ω(√n) on the minimum size of monotone dynamos, namely, ones guaranteeing that throughout the game, a white vertex never turns black. In addition, in a number of these models we derive similar tight bounds (of Ω(√n)) on the minimum size of 2-round dynamos, namely, ones that lead to the all-white state in exactly two rounds. Finally, we make some observations concerning the existence of small “drawing sets”, that lead to nearly-balanced final states.