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.     

A sexual affiliation network of swingers,heterosexuals practicing risk behaviours that potentiate the spread of sexually transmitted infections: A two-mode approach

Using the example of the sexual affiliation networks of swingers, this paper examines how the analysis of sexual affiliation networks can contribute to the development of sexually transmitted infection (STI) prevention strategies. Two-mode network methodology and ERGMs are applied to describe the structural composition of the affiliation network and analyse attribute effects. Swingers were found to recruit their sex partners through one large, moderately cohesive network component. Swingers who used drugs or had a longer history of swinging tended to frequent websites instead of clubs. This study confirms the relevance of studying sexual affiliation networks and its additional value for STI epidemiology.     

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.     

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.     

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.     

Connectedness in a general model for valued networks

Social networks have been closely identified with graph theoretical models, which constitute their most familiar mode of representation. There are a number of such models which may embody symmetric, directed, or valued relationships. But the study of networks with valued linkages, using the natural formalization provided by the valued graph or digraph, has been impeded by a traditional lack of analytical machinery for dealing with valued structures. In this paper, we demonstrate the development and elaboration of formalizations for the central network concepts of reachability, joining, and connectedness through graph theoretical models of increasing complexity, culminating in their expression within a general model for valued structures. This model for valued (symmetric or directed) graphs, or vigraphs, provides a unified representation and matrix methodology for dealing with qualitative and quantitative structures, incorporates many existing methods as special cases, and suggests new applications. Some of the most interesting of these follow the recognition, consistent with the model, that the “values” assigned to network linkages may be sorts of entities other than numbers.     

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.     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time is the class of exponential random graph models (ERGMs), also known as p * models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, Markov chain Monte Carlo (MCMC) algorithms have been developed that produce approximate maximum likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data.
This paper proposes new specifications of exponential random graph models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistics are proposed: geometrically weighted degree distributions, alternating k -triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.     

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.     

Relationship patterns in the 19th century: The friendship network in a German boys’ school class from 1880 to 1881 revisited

The article presents a friendship network from 1880 to 1881 in a school class, which goes back to the exceptional mixed-methods study of the German primary school teacher Johannes Delitsch. The re-analysis of the historic network gives insights into what characteristics defined the friendship networks in school classes in Germany at the end of the 19th century. ERGMs of the so far unmarked data show structural patterns of friendship networks similar to today (reciprocity, transitive triadic closure). Moreover we test the influence of the class ranking order (Lokationsprinzip), which allocates the pupils in the class room according to their school performance. This ranking order produces a hierarchy in the popularity of pupils, through hierarchy–congruent friendship ties going upwards in the hierarchy. In this respect, concerning the effect of school achievement on popularity, we find a strong stratification, which is not always prevalent today.     

On the robustness of centrality measures under conditions of imperfect data

An analysis is conducted on the robustness of measures of centrality in the face of random error in the network data. We use random networks of varying sizes and densities and subject them (separately) to four kinds of random error in varying amounts. The types of error are edge deletion, node deletion, edge addition, and node addition. The results show that the accuracy of centrality measures declines smoothly and predictably with the amount of error. This suggests that, for random networks and random error, we shall be able to construct confidence intervals around centrality scores. In addition, centrality measures were highly similar in their response to error. Dense networks were the most robust in the face of all kinds of error except edge deletion. For edge deletion, sparse networks were more accurately measured.     

Collegial oligarchy and networks of normative alignments in transnational institution building

This paper presents a combined relational and cultural approach to transnational institution building by focusing on a network analysis of a small collegial oligarchy and normative alignments among its peers. To contribute to a theory of institutionalization, we propose hypotheses about whom professionals as institutional entrepreneurs are likely to select as members of their collegial oligarchy, about the role of social networks among them in identifying these leaders, and about the costs of alignments on these leaders’ normative choices. We test these hypotheses using mainly Exponential Random Graph Models (ERGMs) applied to a dataset including network information and normative choices collected at the so-called Venice Forum – a field-configuring event that was central in creating and mobilizing a network of European patent judges for the construction of a new transnational institution, the European Unified Patent Court. We track normative alignments on the collegial hierarchy in this network of judges and their divergent interpretations of the contemporary European patent. Highlighting this under-examined articulation of relational infrastructures and cultural framing in transnational institutionalization shows how Northern European forms of capitalism tend to dominate in this institutionalization process at the expense of the Southern European forms. It also helps reflect on the usefulness of analyses of small networks of powerful players in organizational societies, where power and influence are highly concentrated.     

Notes on blockmodeling

We attempt to develop further the blockmodeling of networks, so as better to capture the network structure. For this purpose a richer structure than ordinary (valued) graphs has to be used for a model. Such structures are valued graphs with typified (complete, dominant, regular, etc.) connections. Based on the proposed formalization, the blockmodeling is cast as an optimization problem.     

The backbone of bipartite projections: Inferring relationships from co-authorship,co-sponsorship,co-attendance and other co-behaviors

The analysis and visualization of weighted networks pose many challenges, which have led to the development of techniques for extracting the network's backbone, a subgraph composed of only the most significant edges. Weighted edges are particularly common in bipartite projections (e.g. networks of co-authorship, co-attendance, co-sponsorship), which are often used as proxies for one-mode networks where direct measurement is impractical or impossible (e.g. networks of collaboration, friendship, alliance). However, extracting the backbone of bipartite projections requires special care. This paper reviews existing methods for extracting the backbone from bipartite projections, and proposes a new method that aims to overcome their limitations. The stochastic degree sequence model (SDSM) involves the construction of empirical edge weight distributions from random bipartite networks with stochastic marginals, and is demonstrated using data on bill sponsorship in the 108th U.S. Senate. The extracted backbone's validity as a network reflecting political alliances and antagonisms is established through comparisons with data on political party affiliations and political ideologies, which offer an empirical ground-truth. The projection and backbone extraction methods discussed in this paper can be performed using the -onemode- command in Stata.     

A deviational approach to blockmodeling of valued networks

This article proposes a novel approach to blockmodeling of valued (one-mode) networks where the identification of (binary) block patterns in the valued relations differ from existing approaches. Rather than looking at the absolute values of relations, or examining valued ties on a per-actor basis (cf. Nordlund, 2007), the approach identifies prominent (binary) ties on the basis of deviations from expected values. By comparing the distribution of each actor's valued relations to its alters with the macro-level distributions of total in- and outdegrees, prominent (1) and non-prominent (0) ties are determined both on a per-actor-to-actor and a per-actor-from-actor basis. This allows for a direct interpretation of the underlying functional anatomy of a non-dichotomized valued network using the standard set of ideal blocks as found in generalized blockmodeling of binary networks.In addition to its applicability for direct blockmodeling, the article also suggests a novel indirect measure of deviational structural equivalence on the basis of such deviations from expected values.Exemplified with the note-sharing data in Žiberna (2007a), citations among social work journals (Baker, 1992), and total commodity trade among EU/EFTA countries as of 2010, both the direct and indirect approach produce results that are more sensitive to variations at the dyadic level than existing approaches. This is particularly evident in the case of the EU/EFTA trade network, where the indirect approach yields partitions and blockmodels in support of theories of regional trade, despite the significantly skewed valued degree distribution of the dataset.     

A Communication Perspective on Video Lottery Terminals

Video Lottery Terminals (VLT) are associated with pathological gambling and with most of the requests for help in combating gambling addiction. Embeddedness of a person in his or her social network is among the communicational factors that may help explain this phenomenon. To verify this, we compared ego networks of VLT gamblers, of gamblers of games with low request for help and of VLT gamblers in treatment (n = 90). The networks of regular VLT gamblers are small and dense and offer little social support. Gamblers in treatment also have small networks, but they are less dense, have more components and offer more social support. Networks of gamblers with low requests for assistance are approximately twice the size as those of VLT gamblers, are sparser and offer more companionship. In conclusion, the VLT gambler is not an isolated individual, but rather an individual ‘shut-in’ a small network of tightly knitted relationships.     

Unequal networks: comparing the pre‐migration overseas networks of Indonesian and Filipino migrant domestic workers

Previous studies of Asian migrant domestic workers' pre‐migration overseas networks have tended to be ethnographic, small‐n case studies such that it is unclear if network differences between migrants are due to individual‐ or country‐level differences or both. This article draws from an original survey of 1,206 Filipino and Indonesian domestic workers in Singapore and Hong Kong to reveal statistically significant differences in the pre‐migration overseas networks of these two nationality groups even after controlling for migrants' educational attainment, marital status, employment status, age, year of first migration, and survey location. Multiple regression analysis highlights how Filipino respondents are more likely than Indonesian respondents to have known existing migrants prior to their first migration from their homeland. Filipino respondents' overseas networks are also significantly larger, more geographically dispersed, and comprise more white‐collar contacts. These findings open up new terrain for migration scholars to study the impact of these nationality‐based network differences on the two groups' divergent migration experiences and aspirations.