SUMMARYThis paper proposes an oscillation model for analyzing the dynamics of activity propagation across social media networks. In order to analyze such dynamics, we generally need to model asymmetric interactions between nodes. In matrix-based network models, asymmetric interaction is frequently modeled by a directed graph expressed as an asymmetric matrix. Unfortunately, the dynamics of an asymmetric matrix-based model is difficult to analyze. This paper, first of all, discusses a symmetric matrix-based model that can describe some types of link asymmetry, and then proposes an oscillation model on networks. Next, the proposed oscillation model is generalized to arbitrary link asymmetry. We describe the outlines of four important research topics derived from the proposed oscillation model. First, we show that the oscillation energy of each node gives a generalized notion of node centrality. Second, we introduce a framework that uses resonance to estimate the natural frequency of networks. Natural frequency is important information for recognizing network structure. Third, by generalizing the oscillation model on directed networks, we create a dynamical model that can describe flaming on social media networks. Finally, we show the fundamental equation of oscillation on networks, which provides an important breakthrough for generalizing the spectral graph theory applicable to directed graphs.

In recent years, with the rapid development of the Internet and cloud computing, an enormous amount of information is exchanged on various social networking services. In order to handle and maintain such a mountain of information properly by limited resources in the network, it is very important to comprehend the dynamics for propagation of information or activity on the social network. One of many indices used by social network analysis which investigates the network structure is "node centrality". A common characteristic of conventional node centralities is that it depends on the topological structure of network and the value of node centrality does not change unless the topology changes. The network dynamics is generated by interaction between users whose strength is asymmetric in general. Network structure reflecting the asymmetric interaction between users is modeled by a directed graph, and it is described by an asymmetric matrix in matrix-based network model. In this paper, we showed an oscillation model for describing dynamics on networks generated from a certain kind of asymmetric interaction between nodes by using a symmetric matrix. Moreover, we propose a new extended index of well-known two node centralities based on the oscillation model. In addition, we show that the proposed index can describe various aspect of node centrality that considers not only the topological structure of the network, but also asymmetry of links, the distribution of source node of activity, and temporal evolution of activity propagation by properly assigning the weight of each link. The proposed model is regarded as the fundamental framework for different node centralities.

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