Oscillation Model for Describing Network Dynamics Caused by Asymmetric Node Interaction

Abstract

SUMMARYThis paper proposes an oscillation model for analyzing the dynamics of activity propagation across social media networks. In order to analyze such dynamics, we generally need to model asymmetric interactions between nodes. In matrix-based network models, asymmetric interaction is frequently modeled by a directed graph expressed as an asymmetric matrix. Unfortunately, the dynamics of an asymmetric matrix-based model is difficult to analyze. This paper, first of all, discusses a symmetric matrix-based model that can describe some types of link asymmetry, and then proposes an oscillation model on networks. Next, the proposed oscillation model is generalized to arbitrary link asymmetry. We describe the outlines of four important research topics derived from the proposed oscillation model. First, we show that the oscillation energy of each node gives a generalized notion of node centrality. Second, we introduce a framework that uses resonance to estimate the natural frequency of networks. Natural frequency is important information for recognizing network structure. Third, by generalizing the oscillation model on directed networks, we create a dynamical model that can describe flaming on social media networks. Finally, we show the fundamental equation of oscillation on networks, which provides an important breakthrough for generalizing the spectral graph theory applicable to directed graphs.

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

“…It is worth to note that this equation has essentially the same structure as the Dirac equation found in relativistic quantum theory [8], [14]. (8) and (12) are mathematically equivalent and they can be transformed into each other by using a simple linear transformation.…”

“…In addition, we choose the eigenvectors v µ of S 0 associated with λ µ in such a way that the eigenvectors form an orthonormal eigenbasis, v µ · v ν = δ µν . Next, we consider a simple and universal interaction model among users through recourse to the concept of the minimal model [8]. We assume that the state of node i at time t (representing a user in the online social network) can be described by a one-dimensional parameter, x i (t).…”

“…This problem is discussed later. Next, let us consider the possibility of the product-form solution of ψ ± (t) for the wave equation (8). The goal here is to write solution ψ ± (t) in product-form, i.e.…”

“…This paper examines a fundamental equation [8] of the oscillation model on networks. The fundamental equation can explicitly describe the causal relation of the influence of the network structure on user dynamics.…”

Dynamical Model of Flaming Phenomena in On-Line Social Networks

“…Oscillation dynamics are quite different from the above topics. Recently, we introduced an oscillation model on networks and investigated its properties [9], [10], [11], [12], [13]. The significance of our oscillation model is that the oscillation energy of each node gives a generalized node centrality that describes the strength of the node's activity.…”