Advances in exponential random graph (p*) models applied to a large social network

Abstract

Recent advances in statistical network analysis based on the family of exponential random graph (ERG) models have greatly improved our ability to conduct inference on dependence in large social networks (Snijders 2002, Pattison and Robins 2002, Handcock 2002, Handcock 2003, Snijders et al. 2006, previous papers this issue). This paper applies advances in both model parameterizations and computational algorithms to an examination of the structure observed in an adolescent friendship network of 1,681 actors from the National Longitudinal Study of Adolescent Health (AddHealth). ERG models of social network structure are fit using the R package statnet, and their adequacy assessed through comparison of model predictions with the observed data for higher-order network statistics.For this friendship network, the commonly used model of Markov dependence leads to the problems of degeneracy discussed by Handcock (2002Handcock ( , 2003. On the other hand, model parameterizations introduced by Snijders et al (2006) and Hunter and Handcock (2006) avoid degeneracy and provide reasonable fit to the data. Degree-only models did a poor job of capturing observed network structure; those that did best included terms both for heterogeneous mixing on exogenous attributes (grade and self-reported race) as well as endogenous clustering. Networks simulated from this model were largely consistent with the observed network on multiple higher-order network statistics, including the number of triangles, the size of the largest component, the overall reachability, the distribution of geodesic distances, the degree distribution, and the shared partner distribution. The ability to fit such models to large datasets and to make inference about the underling processes generating the network represents a major advance in the field of statistical network analysis.The exponential random graph (ERG) class was first posited as an approach to model social network structure almost two decades ago (Frank and Strauss 1986), based upon work in spatial statistics (Besag 1974). The modeling class is extremely general, and as such should in theory be capable of capturing the structure of a wide array of empirical networks, allowing for statistical inference about that structure to be conducted. Nevertheless, most work in this field has focused on a small set of model specifications, most commonly the Markov graphs of Frank and Strauss (1986). Recent work has shown that these commonly used model specifications are in fact not well-suited to capturing the processes underlying many empirical networks, due to the problem of model degeneracy (Handcock 2002(Handcock , 2003. Degeneracy can be described in brief as the phenomenon in which a seemingly reasonable model can actually be such a bad mis-specification for an observed dataset as to render the observed data virtually impossible under the model. Instead, the social processes encapsulated by the model yield networks such as the full or empty graph, which are qualitatively dissimilar to the observed ...

Networks in Public Administration: Current scholarship in review

“…Advances on several fronts have created a sophisticated set of theories and analytical tools that have recently culminated in a 'new science' of networks that spans sociology, physics and organizational sciences (Barabási, 2002;Borgatti et al, 2009;Newman, 2003;Parkhe et al, 2006;Watts, 2004). Statisticians have laid the groundwork for models that allow for inferential hypothesis testing of social theories within a network context as well as longitudinal data analysis to observe and analyse evolution of networks over time (Goodreau, 2007;Handcock, 2003;Krackhardt, 1988;Robins et al, 2007;Snijders, 2002;Snijders et al, 2007).…”

Preference effects on friendship choice: Evidence from an online field experiment

“…Because of this potential confounding, past research has been careful in controlling for structural constraints in studying friendship choice. For instance, researchers have adapted dyad analysis to eliminate the confounding effect of group size (Hallinan and Teixeira, 1987;Moody, 2001;Quillian and Campbell, 2003;Mouw and Entwisle, 2006); controlled individual-level structural variations, such as shared school activities (Moody, 2001) and school segregation (Mouw and Entwisle, 2006); and used exponential random graph models to take balancing reciprocity into consideration (Goodreau, 2007, Goodreau et al, 2009, Wimmer and Lewis, 2010. Furthermore, separating out the effects of structural constraints has been discussed extensively as a methodological challenge (Cheng and Xie, 2012;Currarini et al, 2010;Feld, 1981;McPherson and Smith-Lovin, 1987;Hallinan and Williams, 1989;McPherson et al, 2001;Moody, 2001;Mayer and Puller, 2008;Quillan and Campbell, 2003;Mouw and Entwisle, 2006;Wimmer and Lewis, 2010;Zeng and Xie, 2008).…”

“…We may view this tendency to balance reciprocity in a triadic friendship network as a form of varying exposure, as friendship networks in a closed system cause different dyads to have different levels of exposure, or different potential likelihoods of friendships being formed in the absence of personal preference, due to the existence of common friends. Because prior friendships may be driven at least in part by preference, this raises the possibility that social constraints may be endogenous with respect to preference so as to compound the effects of preference on friendship choice (Goodreau, 2007;Goodreau et al, 2009;Wimmer and Lewis, 2010).…”

“…Furthermore, the balance reciprocity tendency in a triadic friendship network also changes individuals' exposure for friendship purely due to the existence of common friends (Goodreau, 2007;Goodreau et al, 2009;Wimmer and Lewis, 2010). To eliminate the confounding influence of exposure, we designed a field experiment in which we equalized exposure across all participants and eliminated the possibility of balance reciprocity.…”

The Influence of Friendship Network on Graduate Student’s Research Potential

“…These random networks are then compared to the observed networks to assess the likelihood of fit. [10] We start the formal analyses with a baseline model including just the number of edges. Table 1 lists the interesting structure parameters, which are micro-level mechanisms that produce a higher-level network structure.…”