In this paper, we introduce a hierarchical extension of the stochastic blockmodel to identify multilevel community structures in networks. We also present a Markov chain Monte Carlo (MCMC) and a variational Bayes algorithm to fit the model and obtain approximate posterior inference. Through simulated and real datasets, we demonstrate that the model successfully identifies communities and supercommunities when they exist in the data. Additionally, we observe that the model returns a single supercommunity when there is no evidence of multilevel community structure. As expected in the case of the single-level stochastic blockmodel, we observe that the MCMC algorithm consistently outperforms its variational Bayes counterpart. Therefore, we recommend using MCMC whenever the network size allows for computational feasibility.

翻译：本文提出了一种随机块模型的层次扩展方法，用于识别网络中的多级社区结构。我们同时提出了马尔可夫链蒙特卡罗（MCMC）和变分贝叶斯算法来拟合该模型并获取近似的后验推断。通过模拟数据集和真实数据集，我们证明当数据中存在社区和超社区结构时，该模型能够成功识别它们。此外，我们观察到，当数据中没有多级社区结构的证据时，模型会返回一个单一的超社区。正如单层随机块模型情况下的预期，我们观察到MCMC算法在性能上持续优于其变分贝叶斯对应方法。因此，我们建议在网络规模允许计算可行的情况下优先使用MCMC算法。