稠密幂律网络与单纯形复形
Dense Power-law Networks and Simplicial Complexes
摘要：
越来越多的证据表明，密集网络出现在在线社交网络、推荐网络和大脑中。除了密集之外，这些网络通常也是无标度的，即它们的度分布遵循$P(k)\propto k^{-γ}$和$γ\in(1,2]$。生长网络模型已经成功地用于产生使用偏好连接的无标度网络，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恒定的。在这里，我们提出了一个建模框架，产生的网络是密集和无标度。该模型中网络的生长机制基于Pitman-Yor过程。该模型的变体能够产生指数为$γ=2$的无向无标度网络或指数为可调$γ\in (1,2)$的幂律出度分布的有向网络。我们还将模型扩展到有向$2$维单纯复形。单纯形复合体是网络的泛化，它可以编码复杂系统各部分之间的许多身体相互作用，因此越来越受欢迎，以表征从社会交互系统到大脑的不同数据集。我们的模型产生密集的有向单纯复形的幂律分布的广义度的节点。
英文摘要：
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 $P(k)\propto k^{-γ}$ with $γ\in(1,2]$. 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 modelling 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 $γ\in (1,2)$. We also extend the model to that of directed $2$-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.
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