The Majority Rule is applied to a topology that consists of two coupled
random networks, thereby mimicking the modular structure observed in social
networks. We calculate analytically the asymptotic behaviour of the model and
derive a phase diagram that depends on the frequency of random opinion flips
and on the inter-connectivity between the two communities. It is shown that
three regimes may take place: a disordered regime, where no collective
phenomena takes place; a symmetric regime, where the nodes in both communities
reach the same average opinion; an asymmetric regime, where the nodes in each
community reach an opposite average opinion. The transition from the asymmetric
regime to the symmetric regime is shown to be discontinuous.Comment: 14 pages, 4 figure
