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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.
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