Consider a directed network with $K_{r}$ row communities and $K_{c}$ column communities. Previous works found that modeling directed networks in which all nodes have overlapping property requires $K_{r}=K_{c}$ for identifiability. In this paper, we propose an overlapping and nonoverlapping model to study directed networks in which row nodes have overlapping property while column nodes do not. The proposed model is identifiable when $K_{r}\leq K_{c}$. Meanwhile, we provide one identifiable model as extension of ONM to model directed networks with variation in node degree. Two spectral algorithms with theoretical guarantee on consistent estimations are designed to fit the models. A small scale of numerical studies are used to illustrate the algorithms.
翻译：考虑一个与 $K ⁇ r} 列社区 和 $K ⁇ c} 列社区 的定向网络。 先前的工作发现,所有节点具有重叠属性的模拟定向网络需要$K ⁇ r ⁇ K ⁇ c}美元才能识别。 在本文中,我们提出了一个重叠和不重叠的模式,以研究行节点具有重叠属性而列节点不具有重叠属性的定向网络。提议的模型在 $K ⁇ r ⁇ leq K ⁇ c} 美元时可以识别。 同时,我们提供了一个可识别的模型,作为ONM 的扩展模型,以模拟有节点差异的定向网络。 两种具有一致估算理论保证的光谱算法的设计符合模型。 使用小规模的数字研究来说明算法。
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