School of Mathematical Sciences, Queen Mary University of London, E1 4NS, London, United Kingdom.
Phys Rev E. 2018 May;97(5-1):052303. doi: 10.1103/PhysRevE.97.052303.
There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e., their degree distributions follow P(k)∝k^{-γ} with γ∈(1,2]. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time step is constant. Here we present a modeling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent γ=2 or directed networks with power-law out-degree distribution with tunable exponent γ∈(1,2). We also extend the model to that of directed two-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.
越来越多的证据表明，密集网络存在于在线社交网络、推荐网络和大脑中。这些网络不仅密集，而且通常也是无标度的，即它们的度分布遵循 P(k)∝k^{-γ}，其中 γ∈(1,2]。使用优先连接的增长网络模型已经成功地产生了无标度网络，但是这些模型只能产生稀疏网络，因为在每个时间步添加的链接和节点的数量是固定的。在这里，我们提出了一个建模框架，可以产生既密集又无标度的网络。该模型中网络增长的机制基于 Pitman-Yor 过程。该模型的变体能够产生具有指数 γ=2 的无向无标度网络或具有可调指数 γ∈(1,2)的幂律出度分布的有向网络。我们还将模型扩展到有向二维单纯复形。单纯复形是网络的推广，可以编码复杂系统各部分之间的多体相互作用，因此越来越受欢迎，用于描述从社交互动系统到大脑的不同数据集。我们的模型产生了具有幂律分布的节点广义出度的密集有向单纯复形。
