Continuous Time Graph Processes with Known ERGM Equilibria: Contextual Review, Extensions, and Synthesis

Graph processes that unfold in continuous time are of obvious theoretical and practical interest. Particularly useful are those whose long-term behavior converges to a graph distribution of known form. Here, we review some of the conditions for such convergence, and provide examples of novel and/or known processes that do so. These include subfamilies of the well-known stochastic actor oriented models, as well as continuum extensions of temporal and separable temporal exponential family random graph models. We also comment on some related threads in the broader work on network dynamics, which provide additional context for the continuous time case. Graph processes that unfold in continuous time are natural models for social network dynamics: able to directly represent changes in structure as they unfold (rather than, e.g. as snapshots at discrete intervals), such models not only offer the promise of capturing dynamics at high temporal resolution, but are also easily mapped to empirical data without the need to preselect a level of granularity with respect to which the dynamics are defined. Although relatively few general frameworks of this type have been extensively studied, at least one (the stochastic actor-oriented models, or SAOMs) is arguably among the most successful and widely used families of models in the social sciences (see, e.g., Snijders (2001); Steglich et al. (2010); Burk et al. (2007); Sijtsema et al. (2010); de la Haye et al. (2011); Weerman (2011); Schaefer and Kreager (2020) among many others). Work using other continuous time graph processes has also found applications both within (Koskinen and Snijders, 2007; Koskinen et al., 2015; Stadtfeld et al., 2017; Hoffman et al., 2020) and beyond (Grazioli et al., 2019; Yu et al., 2020) the social sciences, suggesting the potential for further advances.

Adolescent Body Mass Index and Exposure to Peers with Overweight and Obesity: A Structural Equation Model Approach to Longitudinal Network Data

Purpose: Considerable evidence has shown that social networks influence a wide variety of health behaviors. This study investigates whether having friends with overweight/obesity in one's social network (network exposure) can predict changes in body mass index (BMI) throughout high school in a diverse urban population of students. Methods: Racially and ethnically diverse students from five high schools in Los Angeles County were surveyed at four time points throughout high school from 2010 to 2013 (N = 2091). Surveys included questions on students' social networks, demographics, and health-related information. BMI and weight categories were calculated for all students who provided height and weight information (∼50%). A latent growth curve model was used to assess the growth trajectory of BMI and the time-varying effect of network exposure to friends with overweight/obesity while controlling for demographic covariates. Results: Hispanic students had a significantly higher initial BMI compared with non-Hispanic students (p < 0.01). There was a significant positive slope for time on BMI growth (p < 0.01). Greater personal network exposure to friends with overweight/obesity was associated with a significant 0.65-point average increase in BMI (p < 0.05) at the first follow-up time point (T2) and a significant 0.62-point average increase in BMI (p < 0.01) at the last follow-up (T4) while controlling for covariates. Conclusions: Using structural equation modeling to understand the relationship between BMI and social networks, we found that increased network exposure to peers with overweight/obesity is associated with higher individual BMI, demonstrating that friendships may influence adolescents' weight status over time.

Contemporaneous Statistics for Estimation in Stochastic Actor-Oriented Co-evolution Models

Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data, collected by observing a network and a behaviour in a panel design. The parameters of SAOMs are usually estimated by the method of moments (MoM) implemented by a stochastic approximation algorithm, where statistics defining the moment conditions correspond in a natural way to the parameters. Here, we propose to apply the generalized method of moments (GMoM), using more statistics than parameters. We concentrate on statistics depending jointly on the network and the behaviour, because of the importance of their interdependence, and propose to add contemporaneous statistics to the usual cross-lagged statistics. We describe the stochastic algorithm developed to approximate the GMoM solution. A small simulation study supports the greater statistical efficiency of the GMoM estimator compared to the MoM.

Continuous measurement of dynamic classroom social interactions

Human observations can only capture a portion of ongoing classroom social activity, and are not ideal for understanding how children's interactions are spatially structured. Here we demonstrate how social interaction can be investigated by modeling automated continuous measurements of children's location and movement using a commercial system based on radio frequency identification. Continuous location data were obtained from 16 five-year-olds observed during three 1-h classroom free play observations. Illustrative coordinate mapping indicated that boys and girls tended to cluster in different physical locations in the classroom, but there was no suggestion of gender differences in children's velocity (i.e., speed of movement). To detect social interaction, we present the radial distribution function, an index of when children were in social contact at greater than chance levels. Rank-order plots indicated that children were in social contact tens to hundreds of times more with some peers than others. We illustrate the use of social ties (higher than average levels of social contact) to visualize the classroom network. Analysis of the network suggests that transitivity is a potential lens through which to examine male, female, and mixed-sex cliques. The illustrative findings suggest the validity of the new measurement approach by re-examining well-established gender segregation findings from a new perspective.