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In this paper, we propose a multi-dimensional opinion formation model for online social networks. Previous studies have not taken into account the "curse of dimensionality" that comes from the higher-dimensional state space used in multi-dimensional opinion formation; the dimensionality is high as opinions with various types of preferences must be handled simultaneously. Conventionally, the interactions between users are limited to static models in which the topics that users discuss are predetermined and do not change. Our proposed method can avoid the above problems with previous studies by introducing low-dimensional subspaces for interacting user pairs by using word distributed representation. We propose a dynamic model that allows users to discuss various topics at each interaction. Simulations of the proposed model show that discussion focused on a particular topic encourages opinion formation.

The structure of social networks or human relationships is difficult to understand since we cannot observed their links and link weights directly. The network resonance method was proposed to obtain information on the unknown Laplacian matrix representing the social network structure. This method extracts information on the eigenvalues and eigenvectors of the Laplacian matrix by observing user dynamics on social networks. The original Laplacian matrix can be reconstructed if all eigenvalues and eigenvectors are known. However, the network resonance method has a problem: the information available about eigenvectors is limited to the absolute value of each element. Therefore, to determine the Laplacian matrix, it is necessary to determine the signs of each element of all eigenvectors. However, sign determination incurs the computation cost of the order of O(2 n) for each of n eigenvectors. This paper proposes a method to determine the signs of each eigenvector element efficiently. The main idea of the method is to generate n 2 − n different sign determination problems for n eigenvectors and to solve them in parallel. All that is required is to obtain n different eigenvectors determined in the shortest time from the n 2 − n different sign determination problems. Since the ratio of the number of sign determinations completed in the method is 1/(n − 1), its efficiency rises with the number of network users. In addition, simulations on networks generated by the BA model show that proposed method offers sign determination in polynomial time.

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The oscillation model has been proposed as a theoretical framework for describing user dynamics in online social networks. This model can model the user dynamics generated by a particular network structure and allow its causal relationships to be explicitly described. In this paper, by applying perturbation theory to the fundamental equation of the oscillation model, we confirm that we can explicitly trace, at least in principle, the changes in user dynamics associated with changes in the network structure. Specifically, we formulate perturbative expansions up to infinite order, by drawing on inferences from regularities found in perturbative expansions; the accuracy of perturbative expansions of finite order is evaluated by numerical experiments.