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.     

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.     

Clearing the FOG: Fuzzy,overlapping groups for social networks

Humans are well known to belong to many associative groups simultaneously, with various levels of affiliation. However, most group detection algorithms for social networks impose a strict partitioning on nodes, forcing entities to belong to a single group. Link analysis research has produced several methods which detect multiple memberships but force equal membership. This paper extends these approaches by introducing the FOG framework, a stochastic model and group detection algorithm for fuzzy, overlapping groups. We apply our algorithm to both link data and network data, where we use a random walk approach to generate rich links from networks. The results demonstrate that not only can fuzzy groups be located, but also that the strength of membership in a group and the fraction of individuals with exclusive membership are highly informative of emerging group dynamics.     

Information communities: The network structure of communication

This study puts forward a variable clique overlap model for identifying information communities, or potentially overlapping subgroups of network actors among whom reinforced independent links ensure efficient communication. We posit that the average intensity of communication between related individuals in information communities is greater than in other areas of the network. Empirical tests show that the variable clique overlap model is indeed more effective in identifying groups of individuals that have strong internal relationships in communication networks relative to prior cohesive subgroup models; the pathways generated by such an arrangement of connections are particularly robust against disruptions of information transmission. Our findings extend the scope of network closure effects proposed by other researchers working with communication networks using social network methods and approaches, a tradition which emphasizes ties between organizations, groups, individuals, and the external environment.     

The Joiners: Active Voluntary Association Membership in Twenty African Countries

To what extent do sub-Saharan Africans actively participate in voluntary groups, and how does development aid influence this involvement? This paper presents a baseline assessment of membership predictors for secular and religious voluntary groups across 20 sub-Saharan African countries. Using Afrobarometer survey data, I adopt an asset-based resources approach in which development aid is seen to mobilize local resources and increase incentives to join voluntary groups. The theory is tested with multilevel, cross-national logistic regression models, including both individual-level and country-level variables to predict active membership. At the individual level, I find that membership in voluntary groups is most likely among those who are well educated, rural-dwelling, male, black, and middle aged—a reflection of social advantage. However, poorer Africans are also likely to join such groups, as are those with a strong religious affinity. At the national level, development aid is positively tied to voluntary group membership, but democratic and economic progresses have little to no consequence on this behavior.     

Information seeking in secondary schools: A multilevel network approach

In this study, we investigate information seeking interactions in secondary schools from a multilevel network approach. Based on network-related theories, we examine the facilitating role of formal subunits. We apply exponential random graph models for multilevel networks and summarize our findings by using a meta-analysis technique. Our results indicate that formal subunits (e.g. subject departments) can, to some extent, facilitate interactions, in loosely coupled organizations (e.g. secondary schools). Finally, this study shows that a multilevel network approach can provide a more informative representation of information seeking ties in knowledge-intensive organizations.     

Analysing networks of networks

We consider data with multiple observations or reports on a network in the case when these networks themselves are connected through some form of network ties. We could take the example of a cognitive social structure where there is another type of tie connecting the actors that provide the reports; or the study of interpersonal spillover effects from one cultural domain to another facilitated by the social ties. Another example is when the individual semantic structures are represented as semantic networks of a group of actors and connected through these actors’ social ties to constitute knowledge of a social group. How to jointly represent the two types of networks is not trivial as the layers and not the nodes of the layers of the reported networks are coupled through a network on the reports. We propose to transform the different multiple networks using line graphs, where actors are affiliated with ties represented as nodes, and represent the totality of the different types of ties as a multilevel network. This affords studying the associations between the social network and the reports as well as the alignment of the reports to a criterion graph. We illustrate how the procedure can be applied to studying the social construction of knowledge in local flood management groups. Here we use multilevel exponential random graph models but the representation also lends itself to stochastic actor-oriented models, multilevel blockmodels, and any model capable of handling multilevel networks.     

Measuring structural HIV stigma

Despite the progress in pharmaceutical and epidemiological tools for combating HIV spread, HIV stigma remains a significant social barrier impeding treatment and prevention efforts, potentially reducing the effectiveness of interventions to reduce HIV transmission. In this paper, we propose a novel approach to defining and estimating HIV stigmatization through the structure of sexual relations, as opposed to attitudes. We conceptualize structural stigma as arising from two mechanisms: (1) a reduced propensity towards HIV serodiscordant partnerships (exclusion); and (2) a reduced propensity towards partnerships with seroconcordant individuals who themselves have serodiscordant partnerships (ostracism). Both mechanisms can be assessed from observed partnership network data using exponential family random graph models (ERGMs). We demonstrate our approach on a sexual contact network of black men who have sex with men in the South Side of Chicago. We find a tendency for serodiscordant sexual relationships to be suppressed in the network ($\theta$ = −0.69, p $<$ .05), as well as a suppressive tendency for HIV negative YBMSM to have sex with other HIV negative people in serodiscordant relationships ($\theta$ = −0.96, p $<$ .05) suggesting that structural HIV stigma is present in this network. Potential relationships with attitudinal stigma and implications for epidemiological strategies for reducing HIV stigma are discussed.     

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.     

Multilevel embeddedness: The case of the global fisheries governance complex

This paper explores how bilateral and multilateral clustering are embedded in a multilevel system of interdependent networks. We argue that in complex systems in which bilateral and multilateral relations are themselves interrelated, such as global fisheries governance, embeddedness cannot be reduced to unipartite or bipartite clustering but implicates multilevel closure. We elaborate expectations for ties’ multilevel embeddedness based on network theory and substantive considerations and explore them using a multilevel ERGM. We find states’ bilateral ties are embedded in their shared membership in multilateral fisheries agreements, which is itself clustered around foci represented by similar content and treaty secretariats.     

Multilevel network interventions: Goals,actions, and outcomes

COVID-19 has resulted in dramatic and widespread social network interventions across the globe, with public health measures such as distancing and isolation key epidemiological responses to minimize transmission. Because these measures affect social interactions between people, the networked structure of daily lives is changed. Such largescale changes to social structures, present simultaneously across many different societies and touching many different people, give renewed significance to the conceptualization of social network interventions. As social network researchers, we need a framework for understanding and describing network interventions consistent with the COVID-19 experience, one that builds on past work but able to cast interventions across a broad societal framework. In this theoretical paper, we extend the conceptualization of social network interventions in these directions. We follow Valente (2012) with a tripartite categorization of interventions but add a multilevel dimension to capture hierarchical aspects that are a key feature of any society and implicit in any network. This multilevel dimension distinguishes goals, actions, and outcomes at different levels, from individuals to the whole of the society. We illustrate this extended taxonomy with a range of COVID-19 public health measures of different types and at multiple levels, and then show how past network intervention research in other domains can also be framed in this way. We discuss what counts as an effective network, an effective intervention, plausible causality, and careful selection and evaluation, as central to a full theory of network interventions.     

The interplay between formal project memberships and informal advice seeking in knowledge-intensive firms: A multilevel network approach

In this study we investigate the interplay between knowledge workers’ formal project team memberships and their informal interactions from a multilevel network perspective. Conceptualizing knowledge workers’ affiliation with project teams as a membership network and their interactions as an advice network, we discuss how shared project team memberships as well as multiple memberships influence patterns of informal exchange in knowledge-intensive organizations. To empirically determine the impact of formal organization on informal exchange we apply exponential random graph models for multilevel networks to relational data collected on 434 R&D employees working on 218 project teams in a high-tech firm in Germany. Our results show that employees sharing project memberships create advice ties to each other but do not exchange advice reciprocally. In addition, we find a negative relationship between having a high number of project memberships and informally seeking or providing advice.     

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.     

Change we can believe in: Comparing longitudinal network models on consistency,interpretability and predictive power

While several models for analysing longitudinal network data have been proposed, their main differences, especially regarding the treatment of time, have not been discussed extensively in the literature. However, differences in treatment of time strongly impact the conclusions that can be drawn from data. In this article we compare auto-regressive network models using the example of TERGMs – a temporal extensions of ERGMs – and process-based models using SAOMs as an example. We conclude that the TERGM has, in contrast to the ERGM, no consistent interpretation on tie-level probabilities, as well as no consistent interpretation on processes of network change. Further, parameters in the TERGM are strongly dependent on the interval length between two time-points. Neither limitation is true for process-based network models such as the SAOM. Finally, both compared models perform poorly in out-of-sample prediction compared to trivial predictive models.     

AN ECONOMETRIC ANALYSIS OF CHARITABLE GIVING WITH INTERDEPENDENT PREFERENCES

Many economists and social scientists have conjectured that individual gifts to charity may be interdependent. This paper explores empirically how an individual's charitable contributions may be affected by the giving of others in a "reference group" of similar individuals. We find modest evidence of interdependence of preferences through these reference groups, although the aggregate effects are not large. Hence, we conclude that the inferences from standard models, which ignore interdependence of preferences, are not likely to be misleading. ( JEL H31, H41, D12)     

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.     

Variations in network boundary and type: A study of adolescent peer influences

This study compares variation in network boundary and network type on network indicators such as degree and estimates of social influences on adolescent substance use. We compare associations between individual use and peer use of tobacco and alcohol when network boundary (e.g., classroom, entire grade in school, and community) and relational type (elicited by asking whom students: (a) are friends with, (b) admire, (c) think will succeed, (d) would like to have a romantic relationship with, and (e) think are popular) are varied. Additionally, we estimate Exponential Random Graph Models (ERGMs) for 232 networks to obtain a homophily estimate for smoking and drinking. Data were collected from a cross-sectional sample of 1707 adolescents in five high schools in one school district in Los Angeles, CA. Results of logistic regression models show that associations were strongest when the boundary condition was least constrained and that associations were stronger for friendship networks than for other ones. Additionally, ERGM estimations show that grade-level friendship networks returned significant homophily effects more frequently than the classroom networks. This study validates existing theoretical approaches to the network study of social influence as well as ways to estimate them. We recommend researchers use as broad a boundary as possible when collecting network data, but observe that for some research purposes more narrow boundaries may be preferred.     

Estimating the effects of network covariates on subgroup insularity with a hierarchical mixed membership stochastic blockmodel

Social networks describe the relationships and interactions among a group of individuals. In many peer relationships, individuals tend to associate more often with some members than others, forming subgroups or clusters. Subgroup structure varies across networks; subgroups may be insular, appearing distinct and isolated from one another, or subgroups may be so integrated that subgroup structure is not visually apparent, and there are numerous ways of quantifying these types of structures. We propose a new model that relates the amount of subgroup integration to network attributes, building on the mixed membership stochastic blockmodel (Airoldi et al., 2008) and subsequent work by Sweet and Zheng (2017) and Sweet et al. (2014). We explore some of the operating characteristics of this model with simulated data and apply this model to determine the relationship between teachers’ instructional practices and their classrooms’ peer network subgroup structure.