Majority opinion diffusion: when tie-breaking rule matters

Consider a graph G, which represents a social network, and assume that initially each node is either blue or white (corresponding to its opinion on a certain topic). In each round, all nodes simultaneously update their color to the most frequent color in their neighborhood. This is called the Majority Model (MM) if a node keeps its color in case of a tie and the Random Majority Model (RMM) if it chooses blue with probability 1/2 and white otherwise. We study the convergence properties of the above models, including stabilization time, periodicity, and the number of stable configurations. In particular, we prove that the stabilization time in RMM can be exponential in the size of the graph, which is in contrast with the previously known polynomial bound on the stabilization time of MM. We provide some bounds on the minimum size of a winning set, which is a set of nodes whose agreement on a color in the initial coloring enforces the process to end in a coloring where all nodes share that color. Furthermore, we calculate the expected final number of blue nodes for a random initial coloring, where each node is colored blue independently with some fixed probability, on cycle graphs. Finally, we conduct some experiments which complement our theoretical findings and also let us investigate other aspects of the models.