Search

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., Comment: arXiv admin note: substantial text overlap with arXiv:2004.09709

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.

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.

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 construct network structure from such data. However, 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 infer the relationships among Senators serving in the 110th United States Congress, the characters in a famous 18th century Chinese novel, and the distribution of flora in North America., Comment: 26 pages, 7 figures, 7 tables