The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with $K$ communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.

中文翻译：

---

随机块模型的极限谱分布

随机块模型（SBM）是 Erdős-Rényi 图的扩展，在数据分析、图数据和社交网络中恢复社区结构等众多领域都有应用。在本文中，我们考虑正常的中心 SBM 邻接矩阵$K$任意规模的社区。当矩阵的大小趋于无穷大时，我们导出了极限经验谱密度函数的显式公式。我们还通过 Stieltjes 变换和随机矩阵理论获得了此类随机矩阵的算子范数的上限。