Mixed Models and Q-Connectivity Graphs for Children’s Play Networks

Q-connectivity graphs display interactions of a target actor. Graph q in the sequence of Q-connectivity graphs for target actor i contains vertices only for peer actors who interact at least q times with actor i; edges connect the peer actors who interact simultaneously with actor i at least q times. The sequence of graphs is not time-ordered but displays the patterns of interaction strength for the target actor. We propose a two-stage model for data from Q-connectivity graphs. We use fixed effects to model observed characteristics of each actor such as gender, and random effects for each actor to capture the heterogeneity between actors. Our model differs from previous work in social networks since we account for multiple observations over the same actor, as well as an actor assuming the role of both the target actor and peer actor. In the first stage, we model probabilities of interaction between two actors to study dyadic relationships. In the second stage, we model probabilities of at least three actors interacting. The two stages combined provide a model for the edges and vertices in a graph in the sequence of Q-connectivity graphs. We apply the models to data on children's play patterns.