In this paper, we first study the fundamental limit of clustering networks when a multi-layer network is present. Under the mixture multi-layer stochastic block model (MMSBM), we show that the minimax optimal network clustering error rate, which takes an exponential form and is characterized by the Renyi divergence between the edge probability distributions of the component networks. We propose a novel two-stage network clustering method including a tensor-based initialization algorithm involving both node and sample splitting and a refinement procedure by likelihood-based Lloyd algorithm. Network clustering must be accompanied by node community detection. Our proposed algorithm achieves the minimax optimal network clustering error rate and allows extreme network sparsity under MMSBM. Numerical simulations and real data experiments both validate that our method outperforms existing methods. Oftentimes, the edges of networks carry count-type weights. We then extend our methodology and analysis framework to study the minimax optimal clustering error rate for mixture of discrete distributions including Binomial, Poisson, and multi-layer Poisson networks. The minimax optimal clustering error rates in these discrete mixtures all take the same exponential form characterized by the Renyi divergences. These optimal clustering error rates in discrete mixtures can also be achieved by our proposed two-stage clustering algorithm.

翻译：本文首先研究多层网络存在时网络聚类的理论极限。在混合多层随机块模型（MMSBM）下，我们证明最小化最大最优网络聚类误差率呈指数形式，且由各组成网络边概率分布之间的Rényi散度刻画。我们提出一种新颖的两阶段网络聚类方法，包括基于张量初始化的算法（涉及节点与样本拆分）以及基于似然的Lloyd算法精化步骤。网络聚类必须与节点社区检测相结合。本算法在MMSBM下实现了最小化最大最优网络聚类误差率，并允许极端网络稀疏性。数值模拟与真实数据实验均验证了本方法优于现有方法。由于网络边常携带计数型权重，我们进一步将方法论与分析框架扩展至离散分布混合模型（包括二项分布、泊松分布及多层泊松网络）的最小化最大最优聚类误差率研究。这些离散混合模型的最小化最大最优聚类误差率均呈相同指数形式，由Rényi散度刻画，且可通过本文提出的两阶段聚类算法实现最优聚类。