Oscillation Model for Describing Network Dynamics Caused by Asymmetric Node Interaction

This 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. key words: spectral graph theory, coupled oscillators, node centrality, resonance, flaming, quantum theory