The phenomenon known as social polarization, in which a social group splits into two or more groups, can cause division of the society by causing the radicalization of opinions and the spread of misinformation, is particularly significant in online communities. To develop technologies to mitigate the effects of polarization in online social networks, it is necessary to understand the mechanism driving its occurrence. There are some models of social polarization in which network structure and users' opinions change, based on the quantified opinions held by the users of online social networks. However, they are based on the interaction between users connected by online social networks. Current recommendation systems offer information from unknown users who are deemed to have similar interests. We can interpret this situation as being yielded non-local effects brought on by the network system, it is not based on local interactions between users. In this paper, based on the spectral graph theory, which can describe non-local effects in online social networks mathematically, we propose a model of polarization that user behavior and network structure change while influencing each other including non-local effects. We investigate the characteristics of the proposed model. Simultaneously, we propose an index to evaluate the degree of network polarization quantitatively, which is needed for our investigations.

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The explosive user dynamics represented by flaming phenomena in online social networks can sometimes negatively influence lives in the real world. To take measures against online flaming phenomena promptly, it is necessary to model its defining characteristics. Based on the oscillation model that describes user dynamics on networks, previous work has revealed that online flaming arises when some eigenvalues of the matrix expressing network structure are non-real numbers. This paper considers the network resonance driven by periodic external stimuli and proposes a flaming model that posits flaming even if all the matrix's eigenvalues are real numbers. Also, we describe a theoretical framework for observing the omen of online flaming to trigger preventive measures.