Statistical network analysis primarily focuses on inferring the parameters of an observed network. In many applications, especially in the social sciences, the observed data is the groups formed by individual subjects. In these applications, the network is itself a parameter of a statistical model. Zhao and Weko (2019) propose a model-based approach, called the hub model, to infer implicit networks from grouping behavior. The hub model assumes that each member of the group is brought together by a member of the group called the hub. The set of members which can serve as a hub is called the hub set. The hub model belongs to the family of Bernoulli mixture models. Identifiability of Bernoulli mixture model parameters is a notoriously difficult problem. This paper proves identifiability of the hub model parameters and estimation consistency under mild conditions. Furthermore, this paper generalizes the hub model by introducing a model component that allows hubless groups in which individual nodes spontaneously appear independent of any other individual. We refer to this additional component as the null component. The new model bridges the gap between the hub model and the degenerate case of the mixture model -- the Bernoulli product. Identifiability and consistency are also proved for the new model. In addition, a penalized likelihood approach is proposed to estimate the hub set when it is unknown.

翻译：统计网络分析主要关注推断观测网络的参数。在许多应用场景中，尤其是在社会科学领域，观测数据是由个体主体所形成的群体。在这些应用中，网络本身是统计模型的一个参数。Zhao与Weko（2019）提出了一种基于模型的方法（称为中心模型），用于从群体行为中推断隐含网络。中心模型假设群体中的每个成员均由该群体中被称为"中心"的成员聚集而成。可作为中心的成员集合称为中心集。该模型属于伯努利混合模型家族。伯努利混合模型参数的可识别性是一个公认的难题。本文证明了在温和条件下中心模型参数的可识别性与估计相合性。此外，本文通过引入一个允许无中心群体的模型组件来推广中心模型——其中个体节点自发出现，且独立于其他任何个体。我们将这一新增组件称为零成分。新模型弥合了中心模型与混合模型的退化情形（即伯努利乘积）之间的差距。针对新模型，本文同样证明了其可识别性与相合性。此外，针对中心集未知的情形，本文提出了一种惩罚似然方法进行估计。