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.     

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.     

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.     

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.     

A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes

This paper describes an empirical comparison of four specifications of the exponential family of random graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood, Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes. At the level of all classes combined, earlier substantive conclusions were supported by all specifications. However, the different specifications led to different conclusions for individual classes. PL produced unreliable estimates (when ML is regarded as the standard) and had more convergence problems than ML. Furthermore, the estimates of covariate effects were affected considerably by controlling for network structure, although the precise specification of the structural part (Markov or partial conditional dependence) mattered less.     

Duality of departmental specializations and PhD exchange: A Weberian analysis of status in interaction using multilevel exponential random graph models (mERGM)

Weber proposes that lifestyle similarities preserve status by producing interactional closure between status similar actors. I investigate this theory on academic status hierarchies by conceptualizing sub-disciplinary specializations as departmental lifestyles and PhD exchange networks as interdepartmental interactions. Multilevel exponential random graph models (mERGM) reveal that the more specializations departments share, the more likely they are to exchange personnel. On the flip side, departments that do not share specializations are less likely to exchange doctoral candidates. Moreover, shared specializations are key determinants of closure between elite departments. These results support Weber’s theory and suggest that shared specializations preserve existing patterns of inequality between elite and non-elite departments.     

Missing data in networks: exponential random graph (p1) models for networks with non-respondents

Survey studies of complete social networks often involve non-respondents, whereby certain people within the “boundary” of a network do not complete a sociometric questionnaire—either by their own choice or by the design of the study—yet are still nominated by other respondents as network partners. We develop exponential random graph (p¹) models for network data with non-respondents. We model respondents and non-respondents as two different types of nodes, distinguishing ties between respondents from ties that link respondents to non-respondents. Moreover, if we assume that the non-respondents are missing at random, we invoke homogeneity across certain network configurations to infer effects as applicable to the entire set of network actors. Using an example from a well-known network dataset, we show that treating a sizeable proportion of nodes as non-respondents may still result in estimates, and inferences about structural effects, consistent with those for the entire network.If, on the other hand, the principal research focus is on the respondent-only structure, with non-respondents clearly not missing at random, we incorporate the information about ties to non-respondents as exogenous. We illustrate this model with an example of a network within and between organizational departments. Because in this second class of models the number of non-respondents may be large, values of parameter estimates may not be directly comparable to those for models that exclude non-respondents. In the context of discussing recent technical developments in exponential random graph models, we present a heuristic method based on pseudo-likelihood estimation to infer whether certain structural effects may contribute substantially to the predictive capacity of a model, thereby enabling comparisons of important effects between models with differently sized node sets.     

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.     

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.     

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.     

A comparison of estimators for the network autocorrelation model based on observed social networks

Network autocorrelation models (NAMs) are widely used to study a response variable of interest among subjects embedded within a network. Although the NAM is highly useful for studying such networked observational units, several simulation studies have raised concerns about point estimation. Specifically, these studies have consistently demonstrated a negative bias of maximum likelihood estimators (MLEs) of the network effect parameter. However, in order to gain a practical understanding of point estimation in the NAM, these findings need to be expanded in three important ways. First, these simulation studies are based on relatively simple network generative models rather than observed networks, thereby leaving as an open question how realistic network topologies may affect point estimation in practice. Second, although there has been strong work done in developing two-stage least squares estimators as well as Bayesian estimators, only the MLE has received extensive attention in the literature, thus leaving practitioners in question as to best practices. Third, the performance of these estimators need to be compared using both bias and variance, as well as the coverage rate of each estimator's corresponding confidence or credible interval. In this paper we describe a simulation study which aims to overcome these shortcomings in the following way. We first fit real social networks using the exponential random graph model and used the Bayesian predictive posterior distribution to generate networks with realistic topologies. We then compared the performance of the three different estimators mentioned above.     

Uncovering CSR meaning networks: A cross-national comparison of Turkey and Slovenia

This study addressed an important question about the meaning of corporate social responsibility (CSR), and how it is measured. Based on a comparison of the meaning networks of CSR in two countries with fundamentally different cultural and socioeconomic backgrounds, we argue that there is a need for an institutional perspective when studying CSR associations and expectations in a particular society. Thus empirical study involved the use of three methods the word-association technique, social network analysis, and blockmodeling using Pajek software; to provide deep insight into the structure of CSR associations. The findings suggest that the two societies have diverse collective cognitive structures regarding CSR. In Turkey, the philanthropic understanding of CSR is highly dominant, while the Slovenian social meaning of CSR is multidimensional. The findings point to the social construction of the concept of CSR with implications both for academic research and practice.     

Are larger cities more central in urban networks: A meta-analysis

As cities develop more and longer-range external relations, some have challenged the long-standing notion that population size indicates a city's power in its urban system. But are population size and network centrality really independent properties in practice, or do larger cities tend to be more central in urban networks? To answer this question, we conducted a systematic literature search and meta-analysed 41 reported correlations between city size and degree centrality. The results show that population size and degree centrality are significantly and positively correlated for cities across various urban systems (r = 0.77), but the correlation varies by network scale and type. The size-centrality association is weaker for global economic and transportation networks (r = 0.43), and stronger for non-global social and communication networks (r = 0.91). This clarifies seemingly contradictory predictions in the literature regarding the association betweensize and centrality for cities.     

Random sampling of alters from networks: A promising direction in egocentric network research

The social network perspective has great potential for advancing knowledge of social mechanisms in many fields. However, collecting egocentric (i.e., personal) network data is costly and places a heavy burden on respondents. This is especially true of the task used to elicit information on ties between network members (i.e., alter-alter ties or density matrix), which grows exponentially in length as network size increases. While most existing national surveys circumvent this problem by capping the number of network members that can be named, this strategy has major limitations. Here, we apply random sampling of network members to reduce cost, respondent burden, and error in network studies. We examine the effectiveness and reliability of random sampling in simulated and real-world egocentric network data. We find that in estimating sample/population means of network measures, randomly selecting a small number of network members produces only minor errors, regardless of true network size. For studies that use network measures in regressions, randomly selecting the mean number of network members (e.g., randomly selecting 10 alters when mean network size is 10) is enough to recover estimates of network measures that correlate close to 1 with those of the full sample. We conclude with recommendations for best practices that will make this versatile but resource intensive methodology accessible to a wider group of researchers without sacrificing data quality.     

Personal support networks of immigrants to Spain: A multilevel analysis

Immigrant flows to Spain have increased greatly in the last decade, but little is known about the composition and role of their personal support networks. Our research questions are: (1) Which factors are associated with ties between immigrants and ‘Spaniards’ (the more settled resident Spanish population), compared with immigrants and non-Spaniards (other immigrants)? (2) Do the support roles of Spaniards and non-Spaniards differ? We analyse personal network (ego-net) survey data. Multilevel logistic regression models are applied, in which the unit of analysis is the undirected tie between an immigrant (ego) and an alter and the dependent variable is whether this tie is to a Spaniard alter, as opposed to a non-Spaniard. We determine the characteristics that are most strongly associated with the probability of a tie between an immigrant and a Spaniard, compared with a non-Spaniard, and consider characteristics of the immigrants (ego), the alters, the relative characteristics of ego-alter, support roles, and local geographical factors. We find a tie to a Spaniard alter is more likely if the immigrant’s country of birth is Portugal or Eastern Europe; if the alter is a work colleague or neighbour; if alter is older than ego. There is geographical variation in the probability of ties to Spaniards, partly explained by the local area presence of co-nationals from the same country of origin as the immigrant. A tie to a Spaniard alter is less likely for immigrants from North Africa (Maghreb); those with no previous contact with Spain; those who are not the first of their peer group/family to immigrate; if ego and alter both work in agriculture. Material help is more likely to be exchanged with a Spaniard alter. Non-Spaniard alters are more likely to exchange help with accommodation and information. ‘Finding a job’ is equally associated with Spaniard and non-Spaniard alters. A tentative conclusion is that some combinations of these characteristics, where a tie to a Spaniard is less likely, may be associated with higher levels of prejudice. Conversely, those characteristics that are positively associated with a tie to a Spaniard may indicate situations where integration of the immigrant population with Spaniards is successfully taking place, and where prejudices are lower, or non-existent. These findings may therefore be helpful for targeting resources to reduce such prejudices. The different types of support exchanged between immigrants and Spaniards and immigrants and non-Spaniards, may indicate current shortfalls in this process, as well as indicating where this support is successfully being exchanged.     

Stakeholder influence,government power,and new opportunities for Chinese Internet companies: A multilevel network analysis of cross-sector networks

In recent decades, Chinese Internet companies have experienced exponential growth. As the Internet industry increasingly commends tremendous financial resources, they also face growing stakeholder expectations for corporate social responsibility (CSR) actions. One way through which Chinese Internet companies conduct CSR is by building cross-sectoral collaborations with nonprofit and nongovernmental organizations (NGOs) and governmental agencies. Aiming to understand Internet companies’ strategic relationship building on CSR issues, the researchers drew from stakeholder influence theory and research on a network approach to stakeholder influence, and applied multilevel network analysis to model three networks related to Chinese Internet companies’ CSR collaborations. Specifically, we found that power and urgency are significant predictors of the structure of Internet companies’ cross-sector CSR alliance network. Organizations affiliated or endorsed by the central Chinese government are the most desirable CSR stakeholders. Additionally, the study also revealed that for Internet companies, devoting their attention to Internet-related social issues could increase their desirability as strategic stakeholders from other sectors and among Internet companies.     

A comparison of simple score and latent class approaches: application to HIV knowledge data in Chinese and multi-country contexts

Knowledge about different health-related attitudes, beliefs, and risks is of significant interest to scholars in different Social Science disciplines. Usually knowledge is collected in a form of multiple variables and then constructed as a composite indicator. The question any researcher working with knowledge-related variables faces is: what is the best way to measure and summarise different dimensions of health-related knowledge? The main goal of this paper is to evaluate and compare simple score and latent class approaches to measuring and summarising health-related knowledge using population data on HIV knowledge collected in five selected countries (China, India, Kenya, Malawi, and Ukraine). The advantages and shortcomings of both approaches (simple score and latent class approaches) to measuring and summarising health-related knowledge are evaluated and discussed.     

Adjustment of issue positions based on network strategies in an election campaign: A two-mode network autoregression model with cross-nested random effects

During election campaigns, political parties deliver statements on salient issues in the news media, which are called issue positions. This article conceptualizes issue positions as a valued and longitudinal two-mode network of parties by issues. The network is valued because parties pronounce pro or con positions on issues in more or less extreme ways. It is longitudinal because the media report new statements of parties on issues each new day.