Scale-freeness and Growth Stability of Realistic Network Models

网络在大量的研究领域中随处可见，从细胞中蛋白质之间的相互作用到万维网的结构，这些网络可能是网络生成模型的结果。生成模型的性质在分析特定网络如何形成时非常有用。一个流行的生成模型是Barabasi-Albert（BA）偏好连接模型，它通过添加一个节点并将其连接到网络中已经存在的节点的迭代过程来形成网络，其概率与它们的度成正比。这个模型导致了一个大致遵循幂律的度分布。因此，很容易声称特定网络具有幂律度分布，因为BA模型可以用来描述网络是如何形成的。然而，在真实的网络中，幂律似乎只适用于大部分数据，通常不适用于完整的度集。这一点在右尾尤其明显，在左尾的程度较小。虽然有几篇论文已经研究了使用不同的模型来研究度分布，但极值理论的方法仍然没有得到充分利用，尽管它们似乎非常适合建模度分布的尾部。幂律对于度分布的不充分性意味着原始BA模型对于网络增长的生成机制的不充分性。该项目旨在使用极值理论的方法为度分布找到一个合适的模型，例如混合模型，其右尾为广义帕累托分布的离散化版本，其余度为幂律。可以随时间记录真实网络的模型参数的变化，这通知BA模型的修改和/或扩展，以生成具有所需度分布的网络，使得新模型令人满意地解释真实网络可能如何增长。

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.