On the Fundamental Equation of User Dynamics and the Structure of Online Social Networks

Abstract

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.

Perturbative Expansion of the Fundamental Equation of Online User Dynamics for Describing Changes in Eigenfrequencies

“…( 4)) [1] perturbative expansion for fundamental equation [6] fundamental equation describing user dynamics with causal relationship (Eq. ( 14)) [1], [25] developing perturbative expansion on simple mode-coupled model up to infinite order with hypergeometric functions (Eqs. ( 71)-( 79))…”

“…We solve this problem by introducing the following fundamental equation of the oscillation model [1], [25]:…”

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

“…Moreover, the oscillation model yields fundamental equations that can describe not only user dynamics but also causal relationships between user dynamics and the structure of OSNs [3,4].…”

“…As one of the models created to describe user dynamics in OSNs, the previous study [1,2] proposed an oscillation model based on the wave equation. Additionally, in the framework of the oscillation model, the fundamental equations that can explicitly describe the causal relationship between user dynamics and certain specific network structures were introduced [3,4]. These fundamental equations of OSNs are of two types: one cannot consider the link structure whereas the other can.…”

Polarization Model of Online Social Networks Based on the Concept of Spontaneous Symmetry Breaking

“…In other words, we want to describe the causal relationship between OSN structure and user dynamics. To achieve this, we need to develop a first-order differential equation with respect to time (hereinafter referred to as the fundamental equation) [5], [6]. Not only is the fundamental equation a first-order differential equation, but its solution must also be the solution of the wave equation ( 4).…”