标题： 邻接谱嵌入在一混合隶属随机块模型中的相合性 显示英文标题

标题： Consistency of adjacency spectral embedding for the mixed membership stochastic blockmodel

摘要： 混合隶属随机块模型是一种用于图的统计模型，它通过允许每个节点在每次决定是否形成边时随机选择不同的社区，扩展了随机块模型。 虽然随机块模型的谱分析越来越成熟，但混合隶属情况的理论发展相对较少。 在这里，我们证明了将邻接谱嵌入到$\mathbb{R}^k$中，然后拟合最小体积包含凸$k$-多胞体到$k-1$主成分上，可以一致地估计一个$k$-社区的混合隶属随机块模型。 关键在于识别混合隶属随机块模型与随机点积图之间的直接对应关系，这大大促进了理论分析。 具体来说，利用随机点积图的$2 \rightarrow \infty$范数和中心极限定理分别展示了该过程的一致性并部分修正了偏差。 显示英文摘要

摘要： The mixed membership stochastic blockmodel is a statistical model for a graph, which extends the stochastic blockmodel by allowing every node to randomly choose a different community each time a decision of whether to form an edge is made. Whereas spectral analysis for the stochastic blockmodel is increasingly well established, theory for the mixed membership case is considerably less developed. Here we show that adjacency spectral embedding into $\mathbb{R}^k$, followed by fitting the minimum volume enclosing convex $k$-polytope to the $k-1$ principal components, leads to a consistent estimate of a $k$-community mixed membership stochastic blockmodel. The key is to identify a direct correspondence between the mixed membership stochastic blockmodel and the random dot product graph, which greatly facilitates theoretical analysis. Specifically, a $2 \rightarrow \infty$ norm and central limit theorem for the random dot product graph are exploited to respectively show consistency and partially correct the bias of the procedure.