  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.
