论文标题
空间不均匀随机图中的聚类函数的缩放
Scaling of the clustering function in spatial inhomogeneous random graphs
论文作者
论文摘要
我们考虑具有可集成连接内核的无限空间不均匀随机图模型，该图在现有的空间随机图模型之间很好地插值。关键示例是重量依赖性随机连接模型的版本，无限几何不均匀的随机图和基于年龄的随机连接模型。这些无限模型是相应有限模型的局部限制，请参见\ cite {lwc_sirgs_2020}。对于这些模型，我们以统一的方式将\ emph {local clustering}的缩放标识为不同制度的根度的函数。我们表明，随着插值参数在不同的方向上移动时，缩放率会表现出相变。除了缩放外，我们还确定了聚类函数的领先常数。这使我们能够得出关于\ emph {典型}三角形的几何形状的结论，这些三角形有助于不同制度中的聚类。
We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection model, the infinite geometric inhomogeneous random graph, and the age-based random connection model. These infinite models arise as the local limit of the corresponding finite models, see \cite{LWC_SIRGs_2020}. For these models we identify the scaling of the \emph{local clustering} as a function of the degree of the root in different regimes in a unified way. We show that the scaling exhibits phase transitions as the interpolation parameter moves across different regimes. In addition to the scaling we also identify the leading constants of the clustering function. This allows us to draw conclusions on the geometry of a \emph{typical} triangle contributing to the clustering in the different regimes.
