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

Derivation and Characteristics of Closed-Form Solutions of the Fundamental Equations for Online User Dynamics