Scale-freeness and Growth Stability of Realistic Network Models

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Networks are found everywhere across a huge number of fields of research, from interactions between proteins in a cell to the structure of the World Wide Web, these networks may be the results of network generative models. The nature of the generative model is useful when it comes to analysing how a particular network has formed. One popular generative model is the Barabasi-Albert (BA) preferential attachment model, which forms networks through the iterative process of adding a node and connecting it to the nodes already in the network with probability proportional to their degree. This model leads to a degree distribution that roughly follows the power law. Therefore, it is tempting to claim that a particular network has a power law degree distribution since the BA model could be used to describe how the network may have formed. However, in real networks, the power law seems to only hold for the bulk of the data and is usually inadequate for the full set of degrees. This is particularly evident in the right tail, and to a lesser extent in the left tail. While several papers have investigated using different models for the degree distribution, methods from extreme value theory remain underutilised even though they seem well suited to the problem of modelling the tails of the degree distribution. The inadequacy of the power law for the degree distribution implies the inadequacy of the original BA model for the generating mechanism for the growth of the network. This project aims to find a suitable model for the degree distribution using methods from extreme value theory, such as mixture models with a discretised version of the Generalised Pareto distribution for the right tail, and power laws for the rest of the degrees. Changes in the model parameters for realistic networks can be recorded over time, which informs the modification and/or extension of the BA model to generate a network with the desired degree distribution, such that the new model satisfactorily explains how realistic networks might have grown.