Recent developments in exponential random graph (p*) models for social networks

This article reviews new specifications for exponential random graph models proposed by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology] and demonstrates their improvement over homogeneous Markov random graph models in fitting empirical network data. Not only do the new specifications show improvements in goodness of fit for various data sets, but they also help to avoid the problem of near-degeneracy that often afflicts the fitting of Markov random graph models in practice, particularly to network data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows estimation of parameters of exponential graph models for many (but not all) cases where it is impossible to estimate parameters of homogeneous Markov graph models. The new specifications were used to model a large number of classical small-scale network data sets and showed a dramatically better performance than Markov graph models. We also review three current programs for obtaining maximum likelihood estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with less accurate pseudo-likelihood estimates. Finally, we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance.     

Closure,connectivity and degree distributions: Exponential random graph (p*) models for directed social networks

The new higher order specifications for exponential random graph models introduced by Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology 36, 99–153] exhibit substantial improvements in model fit compared with the commonly used Markov random graph models. Snijders et al., however, concentrated on non-directed graphs, with only limited extensions to directed graphs. In particular, they presented a transitive closure parameter based on path shortening. In this paper, we explain the theoretical and empirical advantages in generalizing to additional closure effects. We propose three new triadic-based parameters to represent different versions of triadic closure: cyclic effects; transitivity based on shared choices of partners; and transitivity based on shared popularity. We interpret the last two effects as forms of structural homophily, where ties emerge because nodes share a form of localized structural equivalence. We show that, for some datasets, the path shortening parameter is insufficient for practical modeling, whereas the structural homophily parameters can produce useful models with distinctive interpretations. We also introduce corresponding lower order effects for multiple two-path connectivity. We show by example that the in- and out-degree distributions may be better modeled when star-based parameters are supplemented with parameters for the number of isolated nodes, sources (nodes with zero in-degrees) and sinks (nodes with zero out-degrees). Inclusion of a Markov mixed star parameter may also help model the correlation between in- and out-degrees. We select some 50 graph features to be investigated in goodness of fit diagnostics, covering a variety of important network properties including density, reciprocity, geodesic distributions, degree distributions, and various forms of closure. As empirical illustrations, we develop models for two sets of organizational network data: a trust network within a training group, and a work difficulty network within a government instrumentality.     

An introduction to exponential random graph (p) models for social networks

This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement.     

Curved Exponential Family Models for Social Networks

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, and alternating k-twopath statistics of Snijders et al (2006) may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.     

Bayesian analysis for partially observed network data,missing ties,attributes and actors

We consider partially observed network data as defined in Handcock and Gile (2010). More specifically we introduce an elaboration of the Bayesian data augmentation scheme of Koskinen et al. (2010) that uses the exchange algorithm (Caimo and Friel, 2011) for inference for the exponential random graph model (ERGM) where tie variables are partly observed. We illustrate the generating of posteriors and unobserved tie-variables with empirical network data where 74% of the tie variables are unobserved under the assumption that some standard assumptions hold true. One of these assumptions is that covariates are fixed and completely observed. A likely scenario is that also covariates might only be partially observed and we propose a further extension of the data augmentation algorithm for missing attributes. We provide an illustrative example of parameter inference with nearly 30% of dyads affected by missing attributes (e.g. homophily effects). The assumption that all actors are known is another assumption that is liable to be violated so that there are “covert actors”. We briefly discuss various aspects of this problem with reference to the Sageman (2004) data set on suspected terrorists. We conclude by identifying some areas in need of further research.     

A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models

The statistical modeling of social network data is difficult due to the complex dependence structure of the tie variables. Statistical exponential families of distributions provide a flexible way to model such dependence. They enable the statistical characteristics of the network to be encapsulated within an exponential family random graph (ERG) model. For a long time, however, likelihood-based estimation was only feasible for ERG models assuming dyad independence. For more realistic and complex models inference has been based on the pseudo-likelihood. Recent advances in computational methods have made likelihood-based inference practical, and comparison of the different estimators possible.     

Categorical attribute based centrality: E–I and G–F centrality

In a paper examining informal networks and organizational crisis, Krackhardt and Stern (1988) proposed a measure assessing the extent to which relations in a network were internal to a group as opposed to external. They called their measure the E–I index. The measure is now in wide use and is implemented in standard network packages such as UCINET ( Borgatti et al., 2002). The measure is based on a partition-based degree centrality measure and as such can be extended to other centrality measures and group level data. We explore extensions to closeness, betweenness and eigenvector centrality, and show how to apply the technique to sets of subgroups that do not form a partition. In addition, the extension to betweenness suggests a linkage to the Gould and Fernandez brokerage measures, which we explore.     

The Emergence and Stability of Groups in Social Networks

An important puzzle in social network research is to explain how macro-level structures emerge from micro-level network processes. Explaining the emergence and stability of structural groups in social networks is particularly difficult for two reasons. First, because groups are characterized both by high connectedness within (group cohesion) and lack of connectedness between them (group boundaries). Second, because a large number of theoretical micro-level network processes contribute to their emergence. We argue that traditional social network theories that are concerned with the evolution of positive relations (forces of attraction) are not sufficient to explain the emergence of groups because they lack mechanisms explaining the emergence of group boundaries. Models that additionally account for the evolution of negative ties (forces of repulsion) may be better suited to explain the emergence and stability of groups. We build a theoretical model and illustrate its usefulness by fitting stochastic actor-oriented models (SAOMs) to empirical data of co-evolving networks of friendship and dislike among 479 secondary-school students. The SAOMs include a number of newly developed effects expressing the co-evolution between positive and negative ties. We then simulate networks from the estimated models to explore the micro-macro link. We find that a model that considers forces of attraction and repulsion simultaneously is better at explaining groups in social networks. In the long run, however, the empirically informed simulations generate networks that are too stylized to be realistic, raising further questions about model degeneracy and time heterogeneity of group processes.     

A comparison of email networks and off-line social networks: A study of a medium-sized bank

Recently there has been a surge in the availability of online data concerning the connections between people, and these online data are now widely used to map the social structure of communities. There has been little research, however, on how these new types of relational data correspond to classical measures of social networks. To fill this gap, we contrast the structure of an email network with the underlying friendship, communication, and advice seeking networks. Our study is an explorative case study of a bank, and our data contains emails among employees and a survey of the ego networks of the employees. Through calculating correlations with QAP standard errors and estimating exponential random graph (ERG) models, we find that although the email network is related to the survey-based social networks, email networks are also significantly different: while off-line social networks are strongly shaped by gender, tenure, and hierarchical boundaries, the role of these boundaries are much weaker in the email network.     

The co-evolution of gossip and friendship in workplace social networks

This study investigates the co-evolution of friendship and gossip in organizations. Two contradicting perspectives are tested. The social capital perspective predicts that friendship causes gossip between employees, defined as informal evaluative talking about absent colleagues. The evolutionary perspective reverses this causality claiming that gossiping facilitates friendship. The data comprises of three observations of a complete organizational network, allowing longitudinal social network analyses. Gossip and friendship are modeled as both explanatory and outcome networks with RSiena. Results support the evolutionary perspective in that gossip between two individuals increases the likelihood of their future friendship formation. However, individuals with disproportionately high gossip activity have fewer friends in the network, suggesting that the use of gossiping to attract friends has a limit.     

Exponential random graph models for multilevel networks

Modern multilevel analysis, whereby outcomes of individuals within groups take into account group membership, has been accompanied by impressive theoretical development (e.g. Kozlowski and Klein, 2000) and sophisticated methodology (e.g. Snijders and Bosker, 2012). But typically the approach assumes that links between groups are non-existent, and interdependence among the individuals derives solely from common group membership. It is not plausible that such groups have no internal structure nor they have no links between each other. Networks provide a more complex representation of interdependence. Drawing on a small but crucial body of existing work, we present a general formulation of a multilevel network structure. We extend exponential random graph models (ERGMs) to multilevel networks, and investigate the properties of the proposed models using simulations which show that even very simple meso effects can create structure at one or both levels. We use an empirical example of a collaboration network about French cancer research elites and their affiliations (0125 and 0120) to demonstrate that a full understanding of the network structure requires the cross-level parameters. We see these as the first steps in a full elaboration for general multilevel network analysis using ERGMs.     

Exploring the Causal Story: Gender, Active Representation, and Bureaucratic Priorities

Using the theoretical framework posited by Keiser et al. (2002),researchers have found active representation for gender in severalpublic organizations (Keiser et al. 2002; Nicholson-Crotty andMeier 2002; Wilkins and Keiser 2006). This article uses individual-leveldata to explore the causal story behind previous aggregate levelfindings that suggest that female child support enforcementsupervisors provide active representation to female clients,who directly benefit from increased child support collections(Wilkins and Keiser 2006). This article tests whether femalechild support enforcement supervisors behave differently thantheir male counterparts. The findings suggest that female supervisorshave different priorities and/or spend their time differentlythan their male counterparts, and these differences may leadto active representation. Using survey data, I examine the individual-levelcausal story behind the relationship between individuals andbureaucratic priorities.     

Stochastic weighted graphs: Flexible model specification and simulation

In most domains of network analysis researchers consider networks that arise in nature with weighted edges. Such networks are routinely dichotomized in the interest of using available methods for statistical inference with networks. The generalized exponential random graph model (GERGM) is a recently proposed method used to simulate and model the edges of a weighted graph. The GERGM specifies a joint distribution for an exponential family of graphs with continuous-valued edge weights. However, current estimation algorithms for the GERGM only allow inference on a restricted family of model specifications. To address this issue, we develop a Metropolis–Hastings method that can be used to estimate any GERGM specification, thereby significantly extending the family of weighted graphs that can be modeled with the GERGM. We show that new flexible model specifications are capable of avoiding likelihood degeneracy and efficiently capturing network structure in applications where such models were not previously available. We demonstrate the utility of this new class of GERGMs through application to two real network data sets, and we further assess the effectiveness of our proposed methodology by simulating non-degenerate model specifications from the well-studied two-stars model. A working R version of the GERGM code is available in the supplement and is incorporated in the GERGM CRAN package.     

A comparative study of social network models: Network evolution models and nodal attribute models

This paper reviews, classifies and compares recent models for social networks that have mainly been published within the physics-oriented complex networks literature. The models fall into two categories: those in which the addition of new links is dependent on the (typically local) network structure (network evolution models, NEMs), and those in which links are generated based only on nodal attributes (nodal attribute models, NAMs). An exponential random graph model (ERGM) with structural dependencies is included for comparison. We fit models from each of these categories to two empirical acquaintance networks with respect to basic network properties. We compare higher order structures in the resulting networks with those in the data, with the aim of determining which models produce the most realistic network structure with respect to degree distributions, assortativity, clustering spectra, geodesic path distributions, and community structure (subgroups with dense internal connections). We find that the nodal attribute models successfully produce assortative networks and very clear community structure. However, they generate unrealistic clustering spectra and peaked degree distributions that do not match empirical data on large social networks. On the other hand, many of the network evolution models produce degree distributions and clustering spectra that agree more closely with data. They also generate assortative networks and community structure, although often not to the same extent as in the data. The ERGM model, which turned out to be near-degenerate in the parameter region best fitting our data, produces the weakest community structure.     

Obesity-related behaviors in adolescent friendship networks

This study examines obesity-related behaviors within adolescent friendship networks, because adolescent peers have been identified as being important determinants of many health behaviors. We applied ERGM selection models for single network observations to determine if close adolescent friends engage in similar behaviors and to explore associations between behavior and popularity. Same-sex friends were found to be similar on measures of organized physical activity in two out of three school-based friendship networks. Female friends were found to engage in similar screen-based behaviors, and male friends tended to be similar in their consumption of high-calorie foods. Popularity (receiving ties) was also associated with some behaviors, although these effects were gender specific and differed across networks.     

Multilevel Models in Family Research: Some Conceptual and Methodological Issues

Examining the impact of context on individual‐level outcomes has become an increasingly common undertaking in the social sciences. The growth in concern for identifying the effects of macrolevel characteristics has generated both theoretical and methodological advancements. In this issue of Journal of Marriage and Family, Butler (2002) researches whether the effect of welfare benefit levels on premarital childbearing varies by context, Hoffmann (2002) researches the effect of context on adolescent drug use, and Simons et al. (2002) examine how the relationship between parenting and child conduct varies by context. These articles are used as a background to discuss important theoretical and methodological issues surrounding the analysis of multilevel data. The authors present a simple analysis of data pertaining to age at first marriage taken from the Panel Study of Income Dynamics and merged with census data to measure contextual effects as a pedagogical device for introducing readers to the benefits of multilevel modeling.     

Analytic strategies for longitudinal networks with missing data

Missing data are often problematic when analyzing complete longitudinal social network data. We review approaches for accommodating missing data when analyzing longitudinal network data with stochastic actor-based models. One common practice is to restrict analyses to participants observed at most or all time points, to achieve model convergence. We propose and evaluate an alternative, more inclusive approach to sub-setting and analyzing longitudinal network data, using data from a school friendship network observed at four waves (N = 694). Compared to standard practices, our approach retained more information from partially observed participants, generated a more representative analytic sample, and led to less biased model estimates for this case study. The implications and potential applications for longitudinal network analysis are discussed.     

Spectral goodness of fit for network models

We introduce a new statistic, ‘spectral goodness of fit’ (SGOF) to measure how well a network model explains the structure of the pattern of ties in an observed network. SGOF provides a measure of fit analogous to the standard R² in linear regression. Additionally, as it takes advantage of the properties of the spectrum of the graph Laplacian, it is suitable for comparing network models of diverse functional forms, including both fitted statistical models and algorithmic generative models of networks. After introducing, defining, and providing guidance for interpreting SGOF, we illustrate the properties of the statistic with a number of examples and comparisons to existing techniques. We show that such a spectral approach to assessing model fit fills gaps left by earlier methods and can be widely applied.     

Adding valued data to social network measures: Does it add to associations with adolescent substance use?

This research examines whether adding information about adolescent friendships (feeling of closeness to friend, visits to friend's home, interaction outside school, and parent involvement in the friendship) to social network measures increases associations with adolescent substance use. In 2002, friendship nominations were obtained from more than 4700 students in grades 6 to 8 in 13 schools. Social network measures weighted and unweighted for the valued information were created from those data and compared. The findings suggest that, although non-valued network measures and ego substance use were associated, the valued network measures did not add to those associations.     

Limited opportunities: Adolescents’ access to social capital in secondary schools in three European countries

While previous research indicates that students benefit from their peers’ resources, little is known about access to social capital in the school context. Therefore, this study examines differential access to social capital – measured by friends’ socioeconomic status (SES), the number of books they have at home, and their reading habits – in secondary schools in Germany, the Netherlands, and Sweden. Relying on a large-scale dataset, I investigate the association between socioeconomic status, minority status, and social capital using complete friendship network information. I argue that social capital access is connected to a two-stage process consisting of school sorting and friendship selection. To differentiate between these two processes, I apply within-between random effects (REWB). The models show that friendship selection is much less relevant for access to social capital than school sorting. Results indicate that while high-SES students have better access to social capital across dimensions, access patterns for minority students are more nuanced.