A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from data.

翻译：一种具有重叠社区的统计网络模型可以通过相互独立的随机图叠加生成，这些随机图具有不同规模。该模型通过节点数量、社区数量以及社区规模与连边概率的联合分布进行参数化。该模型允许存在稀疏参数区域，这些区域具有幂律极限度分布和非零聚类系数。本文提出了由该模型生成的图样本中团与环频率的大规模近似，这些近似适用于具有无界重叠社区数量的参数区域。我们的结果揭示了这些子图频率的增长速率，并表明其理论密度可以从数据中可靠地估计。