We present a dynamical theory of opinion formation that takes explicitly into account the structure of the social network in which individuals are embedded. The theory predicts the evolution of a set of opinions through the social network and establishes the existence of a martingale property, i.e. the expected weighted fraction of the population that holds a given opinion is constant in time. Most importantly, this weighted fraction is not either zero or one, but corresponds to a non- trivial distribution of opinions in the long time limit. This coexistence of opinions within a social network is in agreement with the often observed locality effect, in which an opinion or a fad is localized to given groups without infecting the whole society. We verified these predictions as well as others concerning the fragility of opinions and the importance of highly connected individuals by computer experiments on scale-free networks.
