Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates around a simplex in the first model, whereas it separates into clusters in the second. By adopting the formalism of generalised random dot-product graphs, we demonstrate that both of these models, and different mixing regimes in the case of mixed membership, may be distinguished by the persistent homology of the underlying point distribution in the case of adjacency spectral embedding. Moreover, despite non-identifiability issues, we show that the persistent homology of the support of the distribution and its super-level sets can be consistently estimated. As an application of our consistency results, we provide a topological hypothesis test for distinguishing the standard and mixed membership stochastic block models.
翻译:使用光谱方法嵌入 $\ mathbb{R ⁇ d} 美元时,众所周知,混合成份和标准随机区块模型等大众网络模型显示出独特的几何结构。 由此产生的点云围绕第一个模型的简单x,而将其分为第二模型的组。 通过采用一般随机点产品图的形式主义,我们证明,在混合成份的情况下,这两种模型和不同的混合制度可以区别于相近光谱嵌入情况下对底点分布的持久性同质性。 此外,尽管存在不可识别的问题,但我们表明,对分布支持及其超级级集的持久性同质性可以进行一致估计。 作为我们一致性结果的应用,我们提供了一种地形假设测试,以区分标准成份和混合成份类群块模型。