Online social networks suffer from explosive user dynamics such as flaming that can seriously affect social activities in the real world because the dynamics have growth rates that can overwhelm our rational decision making faculties. Therefore, a deeper understanding of user dynamics in online social networks is a fundamental problem in computer and information science. One of the effective user dynamics models is the networked oscillation model; it uses a second-order differential equation with Laplacian matrix. Although our previous study indicates that the oscillation model provides us with a minimal but effective model of user interactions, there still remains the open problem as to the existence of a first-order fundamental differential equation that respects the structure of the original network. This paper fills in this gap and shows that, by doubling the dimension of the state space, we can explicitly but naturally construct a fundamental equation that fully respects the structure of the original network.