Revealing of the Underlying Mechanism of Different Node Centralities Based on Oscillation Dynamics on Networks

Abstract

In recent years, with the rapid development of the Internet and cloud computing, an enormous amount of information is exchanged on various social networking services. In order to handle and maintain such a mountain of information properly by limited resources in the network, it is very important to comprehend the dynamics for propagation of information or activity on the social network. One of many indices used by social network analysis which investigates the network structure is "node centrality". A common characteristic of conventional node centralities is that it depends on the topological structure of network and the value of node centrality does not change unless the topology changes. The network dynamics is generated by interaction between users whose strength is asymmetric in general. Network structure reflecting the asymmetric interaction between users is modeled by a directed graph, and it is described by an asymmetric matrix in matrix-based network model. In this paper, we showed an oscillation model for describing dynamics on networks generated from a certain kind of asymmetric interaction between nodes by using a symmetric matrix. Moreover, we propose a new extended index of well-known two node centralities based on the oscillation model. In addition, we show that the proposed index can describe various aspect of node centrality that considers not only the topological structure of the network, but also asymmetry of links, the distribution of source node of activity, and temporal evolution of activity propagation by properly assigning the weight of each link. The proposed model is regarded as the fundamental framework for different node centralities.

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. On the other hand, there are two crucial problems with the fundamental equation (12) as listed below.…”

“…Also, by taking the number of routes passing through a link (or the amount of passing traffic) as the link weight, the oscillation energy for each node gives an value related to the betweenness centrality, again for the non-biased usage condition of the network. In particular, the oscillation energy of each node can generalize the node centrality even in various network usage situations such as having a biased usage condition where a specific node is the information source [12].…”

“…Since it is known that such divergence is not generated by symmetrizable directed graphs, the oscillation model on networks gives a model offers a generation mechanism of explosive user dynamics caused by the network structure [12].…”

“…• The solutions of the fundamental equation (12) are also solutions of the original wave equation (2). Unfortunately, the converse is not true, as shown in Sec.…”

“…If we are to claim that the fundamental equation (12) is really fundamental, it should be possible to derive all solutions of the original wave equation (2) from the fundamental equation (12). • The square root matrix H of the Laplacian matrix that is appeared in the fundamental equation (12) does not reflect the structure of social networks described by the Laplacian matrix L. Since it is unacceptable in practice to hypothesize some direct relationships between nodes where links do not exist in the social network structure described by the Laplacian matrix, the Laplacian matrix L and its square root matrix H should have completely identical link structures. Here, we describe the latter problem via an example.…”