Closed-Form Solutions of the Fundamental Equation That Describes User Dynamics in Online 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 closedform 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.

Derivation and characteristics of closed‐form solutions of the fundamental equations for online user dynamics

“…The solution (17) of the fermion-type fundamental equation ( 7) looks very different in form from the solution (5) of the boson-type fundamental equation (3). In this section, we derive the solutions of the original wave equation ( 1) from the closed-form solutions of the two different fundamental equations ( 3) and (7), and verify that both solutions are general solutions of the original wave equation.…”

“…Details are as follows. From the fermion-type fundamental equation (7) and the equations (B1), (B2), (B3), and (B4) for the Kronecker product, we obtain the second derivative of x(t) with respect to time t as follows:…”

“…In this section, after preparing necessary algebraic relations, we derive a closed-form solution of the fermion-type fundamental equation (7).…”

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

“…According to the literature [21], we show the closed-form solution of the fundamental equation (13). Also, we show the closed-form solution of the fundamental equation (6).…”

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

“…The first type of fundamental equation is easy to solve and we can obtain closed-form solutions, but no closed-form solutions of the second type of fundamental equation have been published so far. This paper is an extended vesiion of a conference paper [5] and derives a closed-form solution of the second type of fundamental equation for OSNs. We also show the closed-form solution of the second type of fundamental equation can generate the general solution of the original wave equation.…”