A necessary and sufficient condition for stable polarization

Polarization has received much attention in recent years. Mechanisms of polarization are found and control protocols for depolarization have been proposed. In spite of these, there is no clear definition of the stability of polarization. To this end, we introduce a co-evolutionary model of discrete opinions and directed networks. We propose an explicit definition of stable polarization based on the mean-field equations. We give a necessary and sufficient condition for the stable polarization, provided that the network evolves much faster than the opinions. In addition, we prove that the polarization is not stable if all individuals are allowed to learn all other opinions, which implies that learning the similar is necessary for the stable polarization. Our work is insightful for the depolarization control.