Opinion evolution among friends and foes: The deterministic majority rule

The influence of the social relationships of an individual on the individual's opinions (about a topic, a product, or whatever else) is a well known phenomenon and it has been widely studied. This paper considers a network of positive (i.e. trusting) or negative (distrusting) social relationships where every individual has an initial positive or negative opinion (about a topic, a product, or whatever else) that changes over time, at discrete time-steps, due to the influences each individual gets from its neighbors. Here, the influence of a trusted neighbor is consistent with the neighbor's opinion, while the influence of a distrusted neighbor is opposite to the neighbor's opinion. At any time, an individual gets one of the two opinions if, at that time, the majority of its neighbors influences the individual to behave so. Given an initial opinion configuration for all the individual, natural questions in this setting are if a given arrival opinion configuration or an equilibrium opinion configuration will ever be reached or if all individuals in a given set will ever get the same (positive) opinion at the same time. All such questions will be proved to result in problems that are tractable if all the relationships are symmetric (that is, the underlying signed graph is undirected), and that are PSpace-complete if the underlying graph is directed even when all the relationships are positive.