Coordination processes in complex systems can be related to the problem of collective ordering in networks, many of which have modular organization. Investigating the order-disorder transition for Ising spins on modular random networks, corresponding to consensus formation in society, we observe two distinct phases: (i) ordering within each module at a critical temperature followed by (ii) global ordering at a lower temperature. This indicates polarization of society into groups having contrary opinions can persist indefinitely even when mutual interactions between agents favor consensus.
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