Abstract
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 hub model belongs to the family of Bernoulli mixture models.
Identifiability of parameters is a notoriously difficult problem for Bernoulli
mixture models. 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. Numerical studies are provided to demonstrate
the theoretical results.
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