  We study discrete preference games in heterogeneous social networks. These
games model the interplay between a player's private belief and his/her
publicly stated opinion (which could be different from the player's belief) as
a strategic game in which the players' strategies are the opinions and the cost
of an opinion in a state is a convex combination through a parameter
$\alpha\in[0,1]$ of two factors: the disagreement between the player's opinion
and his/her internal belief and the number of neighbors whose opinions differ
from the one of the player. The parameter $\alpha$ models how stubborn a player
is: players with large $\alpha$ change their opinion only if many neighbors
disagree with his/her belief. We consider social networks that are
heterogeneous in the sense that the parameter $\alpha$ can vary from player to
player.
  We ask if it is possible that the belief shared by the majority of the
players does not coincide with the opinion that is publicly announced by the
majority of the players in an equilibrium state. Our main result is a
characterization of the social networks that admit an initial belief assignment
for which there exists a sequence of best response moves that reach an
equilibrium in which the initial majority is subverted. Our characterization is
effective in the sense that can be tested efficiently and the initial belief
assignment that can be subverted can be computed in time polynomial in the
number of players. Our result is actually stronger as we show that in each
initial belief assignment that can be subverted, subversion is actually
obtained in a very strong way: it only takes one move of a single player, the
swing player, to lead the social network to a point of no return in which any
rational move from any player leads to a subverted majority.
