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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 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 article proves identifiability of the hub model parameters and estimation consistency under mild conditions. Furthermore, this article 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. for this article are available online. [ABSTRACT FROM AUTHOR]

In medical research, economics, and the social sciences data frequently appear as subsets of a set of objects. Over the past century a number of descriptive statistics have been developed to infer network structure from such data. How- ever, these measures lack a generating mechanism that links the inferred network structure to the observed groups. To address this issue, we propose a model-based approach called the Hub Model which assumes that every observed group has a leader and that the leader has brought together the other members of the group. The performance of Hub Models is demonstrated by simulation studies. We apply this model to the characters in a famous 18th century Chinese novel. [ABSTRACT FROM AUTHOR]