This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clustering of the full graph when the cluster sizes are balanced. A new random degree-based node sampling algorithm is presented which significantly improves upon the performance of the clustering algorithm even when clusters are unbalanced. This framework improves the phase transitions for matrix-decomposition-based clustering with regard to computational complexity and minimum cluster size, which are shown to be nearly dimension-free in the low inter-cluster connectivity regime. A third sampling technique is shown to improve balance by randomly sampling nodes based on spatial distribution. We provide analysis and numerical results using a convex clustering algorithm based on matrix completion.

中文翻译：

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使用随机草图进行可扩展且稳健的社区检测

本文探讨和分析了大型部分观察图的无监督聚类。我们提出了一个可扩展且可证明的随机框架，用于对随机块模型生成的图进行聚类。聚类首先应用于与使用随机采样构造的简化图草图相关联的图邻接矩阵的子矩阵。然后，使用基于相关性的检索步骤，根据从草图中提取的集群来推断全图的集群。当集群大小平衡时，均匀随机节点采样被证明可以提高全图集群的计算复杂度。提出了一种新的基于随机度的节点采样算法，即使在集群不平衡的情况下，该算法也显着提高了聚类算法的性能。该框架在计算复杂度和最小集群大小方面改进了基于矩阵分解的集群的相变，这在低集群间连接机制中几乎是无维度的。第三种采样技术显示了通过基于空间分布随机采样节点来改善平衡。我们使用基于矩阵完成的凸聚类算法提供分析和数值结果。