  The oscillation model, based on the wave equation on networks, can describe
user dynamics in online social networks. The fundamental equation of user
dynamics can be introduced into the oscillation model to explicitly describe
the causal relation of user dynamics yielded by certain specific network
structures. Moreover, by considering the sparseness of the link structure of
online social networks, a novel fundamental equation of different forms has
been devised. In this paper, we derive a closed-form solution of the new
fundamental equation. Also, we show that the closed-form solution of the new
fundamental equation can generate the general solution of the original wave
equation and investigate the characteristics of the derived general solution.
