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.     

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.     

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.     

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.     

How social ties transcend class boundaries? Network variability as tool for exploring occupational homophily

The research on social network analysis established the existence of class homophily, the tendency that personal networks are homogeneous in the class sense, as one of the governing patterns. This is explained via two main mechanisms: choice homophily and induced homophily. But the literature focused less on the question how can class boundaries be transgressed and what are the channels of class heterophily. This paper explores class heterophily on Croatian data acquired through position generator, which measures social capital (resources captured in social relations) by exploring the range of different occupational positions which are accessible to an individual (extensity index). Network variability is thereby taken as proxy for class composition of personal networks. The paper concludes that that political participation and sociability enable cross-class ties, since this offers an opportunity to meet and befriend people from all walks of life; and that people on the middle of the social hierarchy have the most diverse social networks. The hypotheses that social mobility can represent a vehicle for class heterophily; and that class heterophily is more pronounced in smaller settlements, where society networks show more overlap between social circles; were confirmed only partially, and require further investigation. These findings concern class boundaries related to the notion of choice homophily. As for induced homophily, the paper concludes that here too the boundaries are not watertight, as cultural omnivores have a wider range of class contacts.     

Avoidance in negative ties: Inhibiting closure,reciprocity, and homophily

Theorising of negative ties has focused on simplex negative tie networks or multiplex signed tie networks. We examine the fundamental differences between positive and negative tie networks measured on the same set of actors. We test six mechanisms of tie formation on face-to-face positive (affect/esteem) and negative (dislike/disesteem) networks of 282 university students. While popularity, activity, and entrainment are present in both networks, closure, reciprocity, and homophily are largely absent from negative tie networks. We argue this arises because avoidance is inherent to negative sentiments. Avoidance reduces information transfer through negative ties and short-circuits cumulative causation.     

Extrapolative simulation of neighborhood networks based on population spatial distribution: Do they predict crime?

Objectives

Previous criminological scholarship has posited that network ties among neighborhood residents may impact crime rates, but has done little to consider the specific ways in which network structure may enhance or inhibit criminal activity. A lack of data on social ties has arguably led to this state of affairs. We propose to avoid this limitation by demonstrating a novel approach of extrapolatively simulating network ties and constructing structural network measures to assess their effect on neighborhood crime rates.

Methods

We first spatially locate the households of a city into their constituent blocks. Then, we employ spatial interaction functions based on prior empirical work and simulate a network of social ties among these residents. From this simulated network, we compute network statistics that more appropriately capture the notions of cohesion and information diffusion that underlie theories of networks and crime.

Results

We show that these network statistics are robust predictors of the levels of crime in five separate cities (above standard controls) at the very micro geographic level of blocks and block groups.

Conclusions

We conclude by considering extensions of the approach that account for homophily in the formation of network ties.     

Keep around,drop, or revise? exploring what becomes of difficult ties in personal networks

Previous studies increasingly recognize the presence and impact of difficult individuals within personal networks. However, current research sheds little light on the turnover, retention, and change in quality of such difficult ties. The current study addresses this gap by focusing on two distinct forms of network change. We examine how role relationships, support exchange, relational homophily, and personal characteristics are associated with these processes. Data are drawn from three waves of the University of California Social Network Study (UCNets), a project containing comprehensive longitudinal data on ego networks among two adult cohorts. Findings indicate that over time, 34% of difficult ties re-appear in people’s networks as sources of aggravation; 26% of difficult ties are removed from the network; and 40% can no longer be verified as problematic network members. Most ties no longer deemed difficult are identified as providing one or more supportive functions. Multinomial multilevel models reveal that exchanging support tends to anchor difficult ties in the network. Kinship, meanwhile, plays a large role in whether difficult ties remain or get dropped. Personal characteristics such as gender, income, and relocation also play a role in these processes. Overall, we conclude that the turnover dynamics of difficult ties are similar to other ties, and the apparent change from negative to positive suggests that many ties hold ambivalent properties that make shedding such ties less common than may be expected.     

Diffusion in small worlds with homophily and social reinforcement: A theoretical model

We introduce homophily in a percolation model of word-of-mouth diffusion in social networks by reorganizing the nodes according to similarity in preferences for adoption of an innovation. Such preferences are described by a “minimum utility requirement” for an agent to adopt. We show that homophily removes the non-linear relation between preferences and diffusion in the standard percolation model with a high diffusion regime (“hit”) and a low diffusion regime (“flop”). Instead, in a model with perfect homophily, the final diffusion scales linearly with individual preferences: all agents who are willing to adopt, do adopt the innovation. We also investigate the combined effect of homophily and social reinforcement in diffusion. Results indicate that social reinforcement renders clustered networks more efficient in terms of diffusion size for network with strong homophily, while the opposite is true for networks without homophily. The simple structure of our model allows to disentangle the effect of social influence, homophily and the network structure on diffusion. However, the controllability of the theoretical structure comes at the expenses of the realism of the model. For this, we discuss possible extensions and empirical applications.     

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.     

The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density

This study shows several ways that formal graph theoretic statements map patterns of network ties into substantive hypotheses about social cohesion. If network cohesion is enhanced by multiple connections between members of a group, for example, then the higher the global minimum of the number of independent paths that connect every pair of nodes in the network, the higher the social cohesion. The cohesiveness of a group is also measured by the extent to which it is not disconnected by removal of 1, 2, 3,..., k actors. Menger's Theorem proves that these two measures are equivalent. Within this graph theoretic framework, we evaluate various concepts of cohesion and establish the validity of a pair of related measures: 1. Connectivity—the minimum number k of its actors whose removal would not allow the group to remain connected or would reduce the group to but a single member—measures the social cohesion of a group at a general level. 2. Conditional density measures cohesion on a finer scale as a proportion of ties beyond that required for connectivity k over the number of ties that would force it to k + 1.
Calibrated for successive values of k, these two measures combine into an aggregate measure of social cohesion, suitable for both small- and large-scale network studies. Using these measures to define the core of a new methodology of cohesive blocking, we offer hypotheses about the consequences of cohesive blocks for social groups and their members, and explore empirical examples that illustrate the significance, theoretical relevance, and predictiveness of cohesive blocking in a variety of substantively important applications in sociology.     

Actors and networks as objects

This paper explores the application of two contemporary computational methods to the development of sociological theory. Specifically, we combine the methods of object-orientation with discrete event simulation. This approach has several advantages for constructing and evaluating dynamic social theories.In object-oriented program design, objects combine and integrate the traditional concepts of data structures and algorithms, the building blocks of structured programming. Algorithms associated with objects are called methods or member functions. Constructing social actors as objects involves defining both their data attributes and the methods associated with these attributes. We also treat a social network as a computational object. It has data types of nodes and ties. As an object, the network must also have methods that add and delete nodes and ties. Once a network exists, we can create other data types and methods that describe and analyze the network. For example, networks have in-degree and out-degree vectors, and measures of hierarchy. In principle, we can create attributes of networks for all of the structural measures we use to describe networks.We use actor and network objects in a discrete event simulation of a process of formation of dominance structures, exploring several dynamic variations of the underlying theoretical model.     

Analysis of contested reports in exchange networks based on actors’ credibility

Exchanges of information, goods, and services are an essential part of human relations. However, a significant number of reported exchange ties tend to be contested: the reports of the sender and the receiver in an exchange do not concur with each other. To accurately understand the exchange ties between actors and the properties of the associated exchange networks, it is important to address such disagreement. Common practices either eliminate the contested reports or symmetrize them. Neither of them is ideal, as both underuse valuable information in the reports. In this paper, we propose new methods for handling contested exchange ties. The key idea is to measure actors’ credibility based on their asymmetric connections. For example, an actor is less credible the more contested ties she or he has. Using the credibility scores thus calculated, we develop two methods for handling contested ties. The first method is deterministic: it takes the report of the more credible actor as a reflection of the true exchange status between two actors. The second method is stochastic: it assumes the true exchange status between two actors is a random draw from their reports with probabilities proportional to their credibility. We illustrate the above methods by analyzing contested reports in cigarette exchange networks among middle school students in China and social and economic exchange networks among rural households in South Africa. The results show that our methods provide more reasonable corrections to contested reports than previous methods.     

Effects of missing data in social networks

We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three principal missing data mechanisms: network boundary specification (non-inclusion of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos E-print Archive as well as random bipartite graphs. The simulation results show that network boundary specification and fixed choice designs can dramatically alter estimates of network-level statistics. The observed clustering and assortativity coefficients are overestimated via omission of affiliations or fixed choice thereof, and underestimated via actor non-response, which results in inflated measurement error. We also find that social networks with multiple interaction contexts may have certain interesting properties due to the presence of overlapping cliques. In particular, assortativity by degree does not necessarily improve network robustness to random omission of nodes as predicted by current theory.     

Incorporating structural stigma into network analysis

A rich literature has explored the modeling of homophily and other forms of nonuniform mixing associated with individual-level covariates within the exponential family random graph (ERGM) framework. Such differential mixing does not fully explain phenomena such as stigma, however, which involve the active maintenance of social boundaries by ostracism of persons with out-group ties. Here, we introduce a new family of statistics that allows for such effects to be captured, making it possible to probe for the potential presence of boundary maintenance above and beyond simple differences in nomination rates. We demonstrate these statistics in the context of gender segregation in a school classroom, and introduce a framework for understanding the associated coefficients via network perturbation.     

Network Instability in Times of Stability

Personal networks undergo change in response to major life course events. Individual, relational, and network characteristics that influence network instability in the absence of a significant life transition/crisis are less understood. We focus on those ties that transition from active to dormant. Because the shift to dormancy is often interpreted as a reduction in support or social capital, it is considered problematic. This study is based on longitudinal survey data of middle‐class adults who did not undergo life changes. Even in this context of relative stability, support networks experience rates of dormancy similar to those observed during periods of major upheaval. Tie dormancy is unrelated to individual characteristics, network size and density, or homophily along dimensions other than sex. Frequency and medium of communication are particularly notable as factors that were not related to tie dormancy. Ties were less likely to become dormant if they were geographically or emotionally close, immediate kin or neighbors, highly supportive, the same sex, or more embedded in the network. These findings provide context for how support networks operate when not buffeted by exogenous forces. They provide a baseline for understanding the impact on networks of transitions, trauma, new media, and difficult life circumstances.     

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.     

Introduction to stochastic actor-based models for network dynamics

Stochastic actor-based models are models for network dynamics that can represent a wide variety of influences on network change, and allow to estimate parameters expressing such influences, and test corresponding hypotheses. The nodes in the network represent social actors, and the collection of ties represents a social relation. The assumptions posit that the network evolves as a stochastic process ‘driven by the actors’, i.e., the model lends itself especially for representing theories about how actors change their outgoing ties. The probabilities of tie changes are in part endogenously determined, i.e., as a function of the current network structure itself, and in part exogenously, as a function of characteristics of the nodes (‘actor covariates’) and of characteristics of pairs of nodes (‘dyadic covariates’). In an extended form, stochastic actor-based models can be used to analyze longitudinal data on social networks jointly with changing attributes of the actors: dynamics of networks and behavior.     

Goal specific social capital and job satisfaction: Effects of different types of networks on instrumental and social aspects of work

This paper addresses the question “To what extent can job satisfaction be explained as the revenue of social capital?” By conceiving someone’s social network as social capital we specify conditions under which social ties do lead to job satisfaction. We inquire into the idea of goal specificity of social capital, which implies that a network with a given structure and content will have different impacts on various aspects of job satisfaction. If the content of the ties and the structure of the network at the job engender material well-being or produce social approval, satisfaction with the corresponding job aspects increases. Data were collected in 1993 using written questionnaires in two Dutch governmental agencies, one with 32 and the other with 44 employees. These workers’ networks were charted using nine name-generating questions.Social capital, it turns out, is not an all-purpose good but one that is goal specific, even within a single domain of life such as work. Three effects stand out: First, the structure of the network and the content of the ties do matter. Networks of strategic, work-related ties promote an employee’s satisfaction with instrumental aspects of the job, like income, security, and career opportunities. Second, closed networks of identity-based solidarity ties improve an employee’s satisfaction with social aspects of the job, like the general social climate at work and cooperation with management and colleagues. Third, a network with a bow–tie structure (i.e., where a focal actor is the link between two or more mutually exclusive cliques) generally has strong negative effects on satisfaction with the social side of the job; although a bow–tie type network of trusting ties does increase satisfaction with the social side. This implies that Krackhardt’s hypothesis on the unpleasant feelings produced by bow–tie type networks has to be specified for the content of the ties that constitute such a network. The most important conclusion of our analysis is that goal specificity of social capital has implications for both structure and content of social networks. Achievement of a particular goal, such as satisfaction at work, requires not only networks of a certain structure or ties with a particular content, but specifically structured networks of ties with a particular content.     

Evolutionary Process of Social Capital Formation through Community Currency Organizations: The Japanese Case

Community currency systems are said to influence the revival of communities by promoting either local economic growth or social capital accumulation. However, no empirical studies have examined the multiple competing mechanisms for providing social support through transactional networks among participants. The current study connects network structural concepts to theories of social capital, transaction costs, homophily, and resource dependency at multiple levels and evaluates transactional relationships for community rebuilding and local economic development. We examine the evolutionary process of dynamic networks among local residents or organization members with network configurations in one of the largest Japanese community currency systems, “Peanuts.” Using longitudinal network data over 12 years for approximately 1400 actors, we conclude that the evolution and achievement of transactional network dynamics and partner selections differ between the two groups of participants: individual members and organization members. We also provide practical implications for sustaining participants’ transactions and commitment.