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.     

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.     

Logit Models for Affiliation Networks

Once confined to networks in which dyads could be reasonably assumed to be independent, the statistical analysis of network data has blossomed in recent years. New modeling and estimation strategies have made it possible to propose and evaluate very complex structures of dependency between and among ties in social networks. These advances have focused exclusively on one-mode networks—that is, networks of direct ties between actors. We generalize these models to affiliation networks, networks in which actors are tied to each other only indirectly through belonging to some group or event. We formulate models that allow us to study the (log) odds of an actor's belonging to an event (or an event including an actor) as a function of properties of the two-mode network of actors' memberships in events. We also provide illustrative analysis of some classic data sets on affiliation networks.     

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.     

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.     

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.     

Collective action and network change

Network models of collective action commonly assume fixed social networks in which ties influence participation through social rewards. This implies that only certain ties are beneficial from the view of individual actors. Accordingly, in this study we allow that actors strategically revise their relations. Moreover, in our model actors also take into account possible network consequences in their participation choices. To handle the interrelatedness of networks and participation, we introduce new equilibrium concepts. Our equilibrium analysis suggests that structures that tend to segregate contributors from free riders are stable, but costless network change only promotes all-or-nothing participation and complete networks.     

Partitioning signed social networks

Structural balance theory has proven useful for delineating the blockmodel structure of signed social networks. Even so, most of the observed signed networks are not perfectly balanced. One possibility for this is that in examining the dynamics underlying the generation of signed social networks, insufficient attention has been given to other processes and features of signed networks. These include: actors who have positive ties to pairs of actors linked by a negative relation or who belong to two mutually hostile subgroups; some actors that are viewed positively across the network despite the presence of negative ties and subsets of actors with negative ties towards each other. We suggest that instead viewing these situations as violations of structural balance, they can be seen as belonging to other relevant processes we call mediation, differential popularity and internal subgroup hostility. Formalizing these ideas leads to the relaxed structural balance blockmodel as a proper generalization of structural balance blockmodels. Some formal properties concerning the relation between these two models are presented along with the properties of the fitting method proposed for the new blockmodel type. The new method is applied to four empirical data sets where improved fits with more nuanced interpretations are obtained.     

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.     

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.     

An agent-based random-utility-maximization model to generate social networks with transitivity in geographic space

Stochastic actor-based approaches receive increasing interest in the generation of social networks for simulation in time and space. Existing models however cannot be readily integrated in agent-based models that assume random-utility-maximizing behavior of agents. We propose an agent-based model to generate social networks explicitly in geographic space which is formulated in the random-utility-maximizing (RUM) framework. The proposed model consists of a friendship formation mechanism and a component to simulate social encounters in a population. We show how transitivity can be incorporated in both components and how the model can be estimated based on data of personal networks using likelihood estimation. In an application to the Swiss context, we demonstrate the estimation and ability of the model to reproduce relevant characteristics of networks, such as geographic proximity, attribute similarity (homophily), size of personal networks (degree distribution) and clustering (transitivity). We conclude that the proposed social-network model fits seamlessly in existing large-scale micro-simulation systems which assume RUM behavior of agents.     

Modelling the evolution of a bipartite network—Peer referral in interlocking directorates

A central part of relational ties between social actors is constituted by shared affiliations and events. The action of joint participation reinforces personal ties between social actors as well as mutually shared values and norms that in turn perpetuate the patterns of social action that define groups. Therefore the study of bipartite networks is central to social science. Furthermore, the dynamics of these processes suggests that bipartite networks should not be considered static structures but rather be studied over time. In order to model the evolution of bipartite networks empirically we introduce a class of models and a Bayesian inference scheme that extends previous stochastic actor-oriented models for unimodal graphs. Contemporary research on interlocking directorates provides an area of research in which it seems reasonable to apply the model. Specifically, we address the question of how tie formation, i.e. director recruitment, contributes to the structural properties of the interlocking directorate network. For boards of directors on the Stockholm stock exchange we propose that a prolific mechanism in tie formation is that of peer referral. The results indicate that such a mechanism is present, generating multiple interlocks between boards.     

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.     

Who are the active and central actors in the ‘rising civil society’ in Mexico?

The debate about the rise of civil society in Mexico suggests that the processes of political and economic liberalization are multiple and uneven and, thus, have different and contradictory effects on different social groups. This study takes such arguments into account and examines the nature of collective identities and social networks that are more likely to be mobilized in the rising civil society. Who, with what types of social networks and identities, are the active actors in this rising civil society in Mexico? This study also attempts to identify the central actors who take an active part in multi-sector coalitions. As such a broad coalition often leaves profound effects on politics and society, it is vital to ask which actors are likely to take an important step toward multi-sector coalition making. Using a catalog of 1797 protest campaigns collected from three Mexican newspapers between 1964 and 2000, event frequency analysis is employed to find active actors and social network analysis – blockmodel method and degree centrality measure – is applied to uncover central actors. The analyses reveal that while workers, peasants, or students continue to be very active, the centrality of these actors in contentious networks and coalitions has not increased. New central actors in the rising civil society turn out to be civic associations and NGOs formed around single issues, such as environment, retirement, and human rights. When a multi-sector coalition occurs in contemporary Mexico, NGOs and civic associations are likely to play a crucial role in it.     

Dynamic network analysis of contact diaries

Analyzing two-mode networks linking actors to events they attend may help to uncover the structure and evolution of social networks. This classic social network insight is particularly valuable in the analysis of data extracted from contact diaries where contact events produce — and at the same time are the product of relations among participants. Contact events may comprise any number of actors meeting at a specific point in time. In this paper we recall the correspondence between two-mode actor–event networks and hypergraphs, and propose relational hyperevent models (RHEM) as a general modeling framework for networks of time-stamped multi-actor events in which the diarist (“ego”) simultaneously meets several of her alters. RHEM can estimate event intensities associated with each possible subset of actors that may jointly participate in events, and test network effects that may be of theoretical or empirical interest. Examples of such effects include preferential attachment, prior shared activity (familiarity), closure, and covariate effects explaining the propensity of actors to co-attend events. Statistical tests of these effects can uncover processes that govern the formation and evolution of informal groups among the diarist’s alters. We illustrate the empirical value of RHEM using data comprising almost 2000 meeting events of former British Prime Minister Margaret Thatcher with her cabinet ministers, transcribed from contact diaries covering her first term in office (1979–1983).     

Centrality and power revisited: actor success in group decision making

For many years, network analysts viewed positional centrality as a source of social power. More recently, laboratory studies of exchange networks have called the centrality–power link into question: under zero-sum exchange conditions, the ability of certain actors to directly exploit others has been found to account for power independent of actors' centrality. But most observers believe that in non-zero-sum communication networks, centrality should positively affect power. In this study we examine the effect of centrality on power in a communication network involving group voting on political issues. Using a model in which actors' votes are determined by the strength of their initial positions and the social pressures to which they are subjected, we conduct computer simulations to examine the extent to which actors in various network positions achieve favorable political outcomes. Our findings indicate that the link between centrality and power is highly contingent on the structure of the network. In networks with a central actor and an odd number of subgroups, central actors fail to dominate. In fact, in these networks, when peripheral actors are able to directly influence one another, the central actor becomes the least powerful in the network. In networks with a central actor and an even number of subgroups, however, the central actor dominates even in situations with connected peripherals. The highly contingent effect of centrality on power accords with the findings of exchange theorists who have studied power under zero-sum conditions. This raises questions about the nature of the distinction between communication and exchange networks.     

Catching up with big fish in the big pond? Multi-level network analysis through linked design

This article contributes to the study of “duality” [Breiger, R., 1974. The duality of persons and groups. Social Forces 53, 181–190] in social life. Our study explores multi-level networks of superposed and partially connected interdependencies, the first being inter-organizational, the second inter-individual. We propose a method of structural linked design as an articulation for these levels. First, we examine separately the complete networks at each level. Second, we combine the two networks in relation to one another using systematic information about the membership of each individual in the first network (inter-individual) to one of the organizations in the second network (inter-organizational), as in bipartite networks. This dual positioning, or the linked design approach, is carried out in an empirical study examining performance variations within the “elite” of French cancer researchers in 1999. By looking at measures of centrality, we identify the actors that these top researchers consider as central or peripheral at the inter-individual level (the big and the little fish among the elite), and the laboratories that the research directors consider as central or peripheral at the inter-organizational level (the big and the little ponds among all the laboratories conducting cancer research in France at that time). In addition to the rather trivial report of the competitive advantage of big fish in big ponds (particularly because of the advantage of size for laboratories in this field), we use measurements of scientific performance to identify “catching up” strategies that the smallest fish use in this system. We suggest that this method offers new insights into the duality and multi-level dimension of complex systems of interdependencies, and also into the ways in which actors manage these interdependencies. We believe that it adds a new dimension to the sociological exploration of the determinants of performance, of meso-level phenomena such as opportunity structures and institutional change, or of macro-level phenomena such as social inequalities.