Closed-Form Solutions of the Fundamental Equation That Describes User Dynamics in Online Social Networks

TL;DR

This paper derives a closed-form solution for a new fundamental equation modeling user dynamics in online social networks, linking it to the wave equation and accounting for network sparsity.

Contribution

It introduces a novel fundamental equation for user dynamics and provides its closed-form solution, connecting it to the original wave equation.

Findings

Closed-form solution of the new fundamental equation derived.

The solution can generate the general solution of the original wave equation.

The model accounts for network sparsity in social networks.

Abstract

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 online social networks, a novel fundamental equation of different form has been devised. In this paper, we derive a closed-form solution of the new fundamental equation. Also, we find the closed-form solution of the new fundamental equation can generate the general solution of the original wave equation.