  We present a dynamical theory of opinion formation that takes explicitly into
account the structure of the social network in which in- dividuals 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.
that 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 co-existence 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 those concerning the fragility of
opinions and the importance of highly connected individuals in opinion
formation, by performing computer experiments on a number of social networks.
