  Social network analysis presupposes that observed social behavior is
influenced by an unobserved network. Traditional approaches to inferring the
latent network use pairwise descriptive statistics that rely on a variety of
measures of co-occurrence. While these techniques have proven useful in a wide
range of applications, the literature does not describe the generating
mechanism of the observed data from the network.
  In a previous article, the authors presented a technique which used a finite
mixture model as the connection between the unobserved network and the observed
social behavior. This model assumed that each group was the result of a star
graph on a subset of the population. Thus, each group was the result of a
leader who selected members of the population to be in the group. They called
these hub models.
  This approach treats the network values as parameters of a model. However,
this leads to a general challenge in estimating parameters which must be
addressed. For small datasets there can be far more parameters to estimate than
there are observations. Under these conditions, the estimated network can be
unstable.
  In this article, we propose a solution which penalizes the number of nodes
which can exert a leadership role. We implement this as a pseudo-Expectation
Maximization algorithm.
  We demonstrate this technique through a series of simulations which show that
when the number of leaders is sparse, parameter estimation is improved.
Further, we apply this technique to a dataset of animal behavior and an example
of recommender systems.
