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

TL;DR

This paper derives a closed-form solution for a new fundamental equation modeling user dynamics in online social networks, linking it to wave equations and analyzing its properties.

Contribution

It introduces a novel fundamental equation considering network sparsity and derives its closed-form solution, connecting it to existing wave equation models.

Findings

Closed-form solution of the new fundamental equation derived.

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

Characteristics of the general solution are investigated.

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 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.