Hierarchical Mixed-Membership Stochastic Blockmodel With Network-Level Covariates

• In Event: 46.028 - Quantitative Methods for the Analysis of Social Network Data in Education

Abstract

We introduce a new statistical model for social networks, the Hierarchical Mixed Membership Stochastic Blockmodel (HMMSBM) with Network-level Covariates. This model accounts for network-level factors that would influence the amount of mixed membership in a network. This model is under the Hierarchical Network Models (HNM) framework which can model multiple networks at the same time.

In the Stochastic Block Model, the probability of a binary tie is determined by the group membership of the two individuals interacting; individuals in the same group are more likely to have a tie than individuals in different groups. There is single probability vector that determines the likelihood of belonging to each group. In the single network Mixed Membership Stochastic Blockmodel (MMSBM), individuals may belong to different groups for each pairwise interaction and each individual has their own, thus determining the amount of mixed membership in a network. The smaller the value of gamma, the more isolated subgroups in a network.

Our new model extends the single network MMSBM for multiple networks such that we substitute gamma with the natural exponential function in which network-level covariates are predictors. We present both a real-world empirical example using cognitive network data among students in elementary school. The data includes information about classroom friendship relationships in 45 1st, 3rd and 5th grade classrooms and teaching practice variables for each classroom. We fit teaching practice variables to all the classroom networks and results show that the more the positive teaching practice, the more united the students in a classroom, i.e., we will see classroom network with less isolated subgroups.

This study is significant because we can capture and model the relationship between teacher’s behavior and the classroom peer relationship. Previous researches have shown that the more united the students in a classroom, the higher academic achievement they obtain. This can provide insights to school management people on how to make students in classroom more bounded to each other.

Authors

• Qiwen Zheng, University of Maryland - College Park

• Tracy Sweet, University of Maryland - College Park