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 Finally, we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance.     

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.     

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.     

Exponential random graph models for little networks

Statistical models for social networks have enabled researchers to study complex social phenomena that give rise to observed patterns of relationships among social actors and to gain a rich understanding of the interdependent nature of social ties and actors. Much of this research has focused on social networks within medium to large social groups. To date, these advances in statistical models for social networks, and in particular, of Exponential-Family Random Graph Models (ERGMS), have rarely been applied to the study of small networks, despite small network data in teams, families, and personal networks being common in many fields. In this paper, we revisit the estimation of ERGMs for small networks and propose using exhaustive enumeration when possible. We developed an R package that implements the estimation of pooled ERGMs for small networks using Maximum Likelihood Estimation (MLE), called “ergmito”. Based on the results of an extensive simulation study to assess the properties of the MLE estimator, we conclude that there are several benefits of direct MLE estimation compared to approximate methods and that this creates opportunities for valuable methodological innovations that can be applied to modeling social networks with ERGMs.     

Multilevel models for social networks: Hierarchical Bayesian approaches to exponential random graph modeling

In many applications, researchers may be interested in studying patterns of dyadic relationships that involve multiple groups, with a focus on modeling the systematic patterns within groups and how these structural patterns differ across groups. A number of different models – many of them potentially quite powerful – have been developed that allow for researchers to study these differences. However, as with any set of models, these are limited in ways that constrain the types of questions researchers may ask, such as those involving the variance in group-wise structural features. In this paper, we demonstrate some of the ways in which multilevel models based on a hierarchical Bayesian approach might be used to further develop and extend existing exponential random graph models to address such constraints. These include random coefficient extensions to the standard ERGM for sets of multiple unconnected or connected networks and examples of multilevel models that allow for the estimation of structural entrainment among connected groups. We demonstrate the application of these models to real-world and simulated data sets.     

Stable exponential random graph models with non-parametric components for large dense networks

Exponential random graph models (ERGM) behave peculiar in large networks with thousand(s) of actors (nodes). Standard models containing 2-star or triangle counts as statistics are often unstable leading to completely full or empty networks. Moreover, numerical methods break down which makes it complicated to apply ERGMs to large networks. In this paper we propose two strategies to circumvent these obstacles. First, we use a subsampling scheme to obtain (conditionally) independent observations for model fitting and secondly, we show how linear statistics (like 2-stars etc.) can be replaced by smooth functional components. These two steps in combination allow to fit stable models to large network data, which is illustrated by a data example including a residual analysis.     

Exponential random graph model specifications for bipartite networks—A dependence hierarchy

In this paper, we review the development of dependence structures for exponential random graph models for bipartite networks, and propose a hierarchy of dependence structures within which different dependence assumptions may be located. Based on this hierarchy, we propose a new set of model specifications by including bipartite graph configurations involving more than four nodes. We discuss the theoretical significance of the various effects that the extended models afford, and illustrate application of this hierarchy of models to several bipartite networks related to the political mobilization in Brazil in the early 1990s (Mische, 2007).     

Bayesian exponential random graph models with nodal random effects

We extend the well-known and widely used exponential random graph model (ERGM) by including nodal random effects to compensate for heterogeneity in the nodes of a network. The Bayesian framework for ERGMs proposed by Caimo and Friel (2011) yields the basis of our modelling algorithm. A central question in network models is the question of model selection and following the Bayesian paradigm we focus on estimating Bayes factors. To do so we develop an approximate but feasible calculation of the Bayes factor which allows one to pursue model selection. Three data examples and a small simulation study illustrate our mixed model approach and the corresponding model selection.     

Advances in Exponential Random Graph (p*) Models Applied to a Large Social Network

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, Hunter et al. 2005, Goodreau et al. 2005, 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 (2002, 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.     

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.     

Efficient Bayesian inference for exponential random graph models by correcting the pseudo-posterior distribution

Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves the calculation of an intractable normalizing constant. This barrier motivates the consideration of tractable approximations to the likelihood function, such as the pseudolikelihood function, which offers an approach to constructing such an approximation. Naive implementation of what we term a pseudo-posterior resulting from replacing the likelihood function in the posterior distribution by the pseudolikelihood is likely to give misleading inferences. We provide practical guidelines to correct a sample from such a pseudo-posterior distribution so that it is approximately distributed from the target posterior distribution and discuss the computational and statistical efficiency that result from this approach. We illustrate our methodology through the analysis of real-world graphs. Comparisons against the approximate exchange algorithm of Caimo and Friel (2011) are provided, followed by concluding remarks.     

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.     

Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models

Social network data often involve transitivity, homophily on observed attributes, community structure, and heterogeneity of actor degrees. We propose a latent cluster random effects model to represent all of these features, and we develop Bayesian inference for it. The model is applicable to both binary and non-binary network data. We illustrate the model using two real datasets: liking between monks and coreaderships between Slovenian publications. We also apply it to two simulated network datasets with very different network structure but the same highly skewed degree sequence generated from a preferential attachment process. One has transitivity and community structure while the other does not. Models based solely on degree distributions, such as scale-free, preferential attachment and power-law models, cannot distinguish between these very different situations, but the latent cluster random effects model does.     

The local and global structure of knowledge production in an emergent research field: An exponential random graph analysis

Previous research has characterized knowledge networks by diffuse connectivity and/or clusters and the absence of centrality. In contrast, exponential random graph models used in this article demonstrate that the uncertainty and centralized influence typical of an emerging area of research leads to the creation of a densely interconnecting core that acts to cohere the network. Moreover, eclecticism and innovativeness, also characteristic of a developing area, lead to a diffusely connected structure. The data, comprising 2200 authors and 76 papers have been manually coded from articles on the feminization of the labor force in Asia.     

Parameter estimation procedures for exponential-family random graph models on count-valued networks: A comparative simulation study

The exponential-family random graph models (ERGMs) have emerged as an important framework for modeling social networks for a wide variety of relational types. ERGMs for valued networks are less well-developed than their unvalued counterparts, and pose particular computational challenges. Network data with edge values on the non-negative integers (count-valued networks) is an important such case, with examples ranging from the magnitude of migration and trade flows between places to the frequency of interactions and encounters between individuals. Here, we propose an efficient parallelizable subsampled maximum pseudo-likelihood estimation (MPLE) scheme for count-valued ERGMs, and compare its performance with existing Contrastive Divergence (CD) and Monte Carlo Maximum Likelihood Estimation (MCMLE) approaches via a simulation study based on migration flow networks in two U.S. states. Our results suggest that edge value variance is a key factor in method performance, while network size mainly influences their relative merits in computational time. For small-variance networks, all methods perform well in point estimations while CD greatly overestimates uncertainties, and MPLE underestimates them for dependence terms; all methods have fast estimation for small networks, but CD and subsampled multi-core MPLE provides speed advantages as network size increases. For large-variance networks, both MPLE and MCMLE offer high-quality estimates of coefficients and their uncertainty, but MPLE is significantly faster than MCMLE; MPLE is also a better seeding method for MCMLE than CD, as the latter makes MCMLE more prone to convergence failure. The study suggests that MCMLE and MPLE should be the default approach to estimate ERGMs for small-variance and large-variance valued networks, respectively. We also offer further suggestions regarding choice of computational method for valued ERGMs based on data structure, available computational resources and analytical goals.     

Conditionally exponential random models for individual properties and network structures: Method and application

Exponential random models have been widely adopted as a general probabilistic framework for complex networks and recently extended to embrace broader statistical settings such as dynamic networks, valued networks or two-mode networks. Our aim is to provide a further step into the generalization of this class of models by considering sample spaces which involve both families of networks and nodal properties verifying combinatorial constraints. We propose a class of probabilistic models for the joint distribution of nodal properties (demographic and behavioral characteristics) and network structures (friendship and professional partnership). It results in a general and flexible modeling framework to account for homophily in social structures. We present a Bayesian estimation method based on the full characterization of their sample spaces by systems of linear constraints. This provides an exact simulation scheme to sample from the likelihood, based on linear programming techniques. After a detailed analysis of the proposed statistical methodology, we illustrate our approach with an empirical analysis of co-authorship of journal articles in the field of neuroscience between 2009 and 2013.     

Multiplexity and strategic alliances: The relational embeddedness of coalitions in social movement organisational fields

While scholars have embraced the notion of social movements as networks, there has been little empirical exploration of the emergence of coalitions within these multilayered systems. Here I explore the role of overlapping relations in alliance formation amongst a group of 55 health-related professional social movement organisations mobilised against austerity. Using cross-sectional bivariate exponential random graph models, I find dependencies between digital proxies for alliance, shared allies, information exchange, positive nomination and offline colobbying activity at the dyadic, degree and triadic levels. Cross-network associations indicate that multiplexity plays a non-trivial role in the formation of alliances and, more generally, social movement organisational fields, necessitating increased attention from scholars of social movements.     

Networks and geography: Modelling community network structures as the outcome of both spatial and network processes

This paper focuses on how to extend the exponential random graph models to take into account the geographical embeddedness of individuals in modelling social networks. We develop a hierarchical set of nested models for spatially embedded social networks, in which, following Butts (2002), an interaction function between tie probability and Euclidean distance between nodes is introduced. The models are illustrated by an empirical example from a study of the role of social networks in understanding spatial clustering in unemployment in Australia. The analysis suggests that a spatial effect cannot solely explain the emergence of organised network structure and it is necessary to include both spatial and endogenous network effects in the model.