A study on stochastic properties of large scale-free networks

Abstract

A scale-free network is the type of network where the edge strength is of no interest but its existence only counts. The network is a useful tool for representing linkages between nodes such as the linkages between participants in a social network and those in a world-wide community or in a internet. A main feature of the scale-free network is the node-degree distribution which is mostly in the form of a power function of the node-degree. As for the BA (Barabasi-Albert) model, the power function is of degree 3. In this thesis, we generalized the BA model in several aspects. One of them is that we introduced uncertainty to the number of the new edges which are created by a new comer to a network, while the number is a constant for the BA model. We investigated the effect of the uncertainty, in particular, in the context of the mean and variance of the uncertainty model. Another is that, when the number of newly created edges is assumed dependent upon the age of the network, we derived exact distributions of the node-degrees. We also investigated the entropy of the node-degree distribution of a network and compared our proposed model with others with a favorable result. As for the initial attractiveness of a node, our generalized model was shown to be a generalized version of the existing results. In a nutshell, our introduction of uncertainty to the number of newly created edges leads us to a generalization of the BA model in several aspects.