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.     

Network centrality and social movement media coverage: A two-mode network analytic approach

This article examines the relationship between structural location (namely, degree centrality) and news media coverage. Our central hypothesis is that the network centrality of social movement actors is positively associated with the prevalence of actors being cited in the print news media. This paper uses two-mode data from a communication network of environmentalists in British Columbia, and examines the relationship between their structural location and the frequency by which they are cited in newsprint media with regard to particular frames (about forest conservation, environmental protest, and related issues). We asked a sample of social movement participants about their ties to a target list of relatively high profile actors (environmental activists). We turned the resulting network matrix into a bipartite graph that examined the relationships amongst the target actors vis a vis the respondents. Next we calculated point in-degree for the target actors. For the target actors we also have data from a representative sample of 957 print news articles about forestry and conservation of old growth forests in British Columbia. We compare the effects of network centrality of the target actor versus several attributes of the target actors (gender, level of radicalism, leadership status) on the amount of media coverage that each of the target actors receives. We find that network centrality is associated with media coverage controlling for actor attributes. We discuss theoretical implications of this research. Finally, we also discuss the methodological pros and cons of using a “target name roster” to construct two-mode data on social movement activists.     

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).     

Stochastic a posteriori blockmodels: Construction and assessment

In 1983, Holland, Laskey, and Leinhardt, using the ideas of Holland and Leinhardt, and Fienberg and Wasserman, introduced the notion of a stochastic blockmodel. The mathematics for stochastic a priori blockmodels, in which exogenous actor attribute data are used to partition actors independently of any statistical analysis of the available relational data, have been refined by several researchers and the resulting models used by many. Attempts to simultaneously partition actors and to perform relational data analyses using statistical methods that yield stochastic a posteriori blockmodels are still quite rare. In this paper, we discuss some old suggestions for producing such posterior blockmodels, and comment on other new suggestions based on multiple comparisons of model parameters, log-linear models for ordinal categorical data, and correspondence analysis. We also review measures for goodness-of-fit of a blockmodel, and we describe a natural approach to this problem using likelihood-ratio statistics generated from a popular model for relational data.     

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.     

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.     

Random errors in egocentric networks

The systematic errors that are induced by a combination of human memory limitations and common survey design and implementation have long been studied in the context of egocentric networks. Despite this, little if any work exists in the area of random error analysis on these same networks; this paper offers a perspective on the effects of random errors on egonet analysis, as well as the effects of using egonet measures as independent predictors in linear models. We explore the effects of false-positive and false-negative error in egocentric networks on both standard network measures and on linear models through simulation analysis on a ground truth egocentric network sample based on facebook-friendships. Results show that 5–20% error rates, which are consistent with error rates known to occur in ego network data, can cause serious misestimation of network properties and regression parameters.     

Social Networks and Social Settings: Developing a Coevolutionary View

One way to think about social context is as a sample of alters. To understand individual action, therefore, it matters greatly where these alters may be coming from, and how they are connected. According to one vision, connections among alters induce local dependencies—emergent rules of social interaction that generate endogenously the observed network structure of social settings. Social selection is the decision of interest in this perspective. According to a second vision, social settings are collections of social foci—physical or symbolic locales where actors meet. Because alters are more likely to be drawn from focused sets, shared social foci are frequently considered as the main generators of network ties, and hence of setting structure. Affiliation to social foci is the decision of central interest in this second view. In this paper we show how stochastic actor–oriented models (SAOMs) originally derived for studying the dynamics of multiple networks may be adopted to represent and examine these interconnected systems of decisions (selection and affiliation) within a unified analytical framework. We illustrate the empirical value of the model in the context of a longitudinal sample of adolescent participating in the Glasgow Teenage Friends and Lifestyle Study. Social selection decisions are examined in the context of networks of friendship relations. The analysis treats musical genres as the main social foci of interest.     

Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models

Social network data often involve transitivity, homophily on observed attributes, community structure, and heterogeneity of actor degrees. We propose a latent cluster random effects model to represent all of these features, and we develop Bayesian inference for it. The model is applicable to both binary and non-binary network data. We illustrate the model using two real datasets: liking between monks and coreaderships between Slovenian publications. We also apply it to two simulated network datasets with very different network structure but the same highly skewed degree sequence generated from a preferential attachment process. One has transitivity and community structure while the other does not. Models based solely on degree distributions, such as scale-free, preferential attachment and power-law models, cannot distinguish between these very different situations, but the latent cluster random effects model does.     

Weight matrices for social influence analysis: An investigation of measurement errors and their effect on model identification and estimation quality

Weight matrices, such as used in network autocorrelation models, are useful to investigate social influence processes. The objective of this paper is to investigate a key topic that has received relatively little attention in previous research, namely the issues that arise when observational limitations lead to measurement errors in these weight matrices. Measurement errors are investigated from two perspectives: when relevant ties are omitted, and when irrelevant ties are erroneously included as part of the matrix. The paper first shows analytically that these two situations result in biased estimates. Next, a simulation experiment provides evidence of the effect of erroneously coding the weight matrix on model performance and the ability of a network autocorrelation test to identify social influence effects. The results suggest that depending on the level of autocorrelation and the topology attributes of the underlying matrix, there is a window of opportunity to identify and model social influence processes even in situations where the ties in a matrix cannot be accurately observed.     

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.     

Testing centralization in random graphs

Centrality is an important concept in social network analysis which involves identification of important or prominent actors. Three common definitions of centrality are degree centrality, closeness centrality and betwenness centrality which yield actor indices. By aggregating these actor indices of centrality across actors, we obtain a single group-level index of centralization. In this paper, we consider the problem of testing whether the observed data is likely to have come from a particular kind of centralized structure of a given size, edge probability and extent of centralization. Eight different group-level indices of centralization are used as test statistics of graph centralization. As our graph model, we assume a general blockmodel which allows a rich probabilistic structure. By carrying out a simulation study the performance of the tests is evaluated by comparing their power functions. The results imply that two tests based on degree and four tests based on closeness have high power. In addition, critical values of the tests are modeled conditional on graph parameters via a linear regression model. An application is illustrated with analysis on a real data set.     

Structural holes and bridging in two-mode networks

Social networks are often structured in such a way that there are gaps, or “structural holes,” between regions. Some actors are in the position to bridge or span these gaps, giving rise to individual advantages relating to brokerage, gatekeeping, access to non-redundant contacts, and control over network flows. The most widely used measures of a given actor’s bridging potential gauge the extent to which that actor is directly connected to others who are otherwise not well connected to each other. Unfortunately, the measures that have been developed to identify structural holes cannot be adapted directly to two-mode networks, like individual-to-organization networks. In two-mode networks, direct contacts cannot be directly connected to each other by definition, making the calculation of redundancy, effective size, and constraint impossible with conventional one-mode methods. We therefore describe a new framework for the measurement of bridging in two-mode networks that hinges on the mathematical concept of the intersection of sets. An actor in a given node class (“ego”) has bridging potential to the extent that s/he is connected to actors in the opposite node class that have unique profiles of connections to actors in ego’s own node class. We review the relevant literature pertaining to structural holes in two-mode networks, and we compare our primary bridging measure (effective size) to measures of bridging that result when using one-mode projections of two-mode data. We demonstrate the results of applying our approach to empirical data on the organizational affiliations of elites in a large U.S. city.     

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.     

A New Model for Information Diffusion in Heterogeneous Social Networks

This paper discusses a new model for the diffusion of information through heterogeneous social networks. In earlier models, when information was given by one actor to another the transmitter did not retain the information. The new model is an improvement on earlier ones because it allows a transmitter of information to retain that information after telling it to somebody else. Consequently, the new model allows more actors to have information during the information diffusion process. The model provides predictions of diffusion times in a given network at the global, dyadic, and individual levels. This leads to straightforward generalizations of network measures, such as closeness centrality and betweenness centrality, for research problems that focus on the efficiency of information transfer in a network. We analyze in detail how information diffusion times and centrality measures depend on a series of network measures, such as degrees and bridges. One important finding is that predictions about the time actors need to spread information in the network differ considerably between the new and old models, while the predictions about the time needed to receive information hardly differ. Finally, some cautionary remarks are made about using the model in empirical research.     

The Network Autocorrelation Model using Two-mode Data: Affiliation Exposure and Potential Bias in the Autocorrelation Parameter

The network autocorrelation model has been a workhorse for modeling network influences on individual behavior. The standard network approaches to mapping social influence using network measures, however, are limited to specifying an influence weight matrix (W) based on a single mode network. Additionally, it has been demonstrated that the estimate of the autocorrelation parameter ρ of the network effect tends to be negatively biased as the density in W matrix increases. The current study introduces a two-mode version of the network autocorrelation model. We then conduct simulations to examine conditions under which bias might exist. We show that the estimate for the affiliation autocorrelation parameter (ρ) tends to be negatively biased as density increases, as in the one-mode case. Inclusion of the diagonal of W, the count of the number of events participated in, as one of the variables in the regression model helps to attenuate such bias, however. We discuss the implications of these results.     

Prismatic effects of negative ties

Research on negative ties has focused primarily on the harm they do. In this paper, we show that negative ties can also have beneficial effects. We argue that, like positive ties, negative ties can link actors together in the minds of observers. As a result, we theorize that negative ties with high-status actors can benefit a focal actor, whereas negative ties with low-status actors can harm the focal actor. This prismatic effect depends on the existing status of the focal actor: a focal actor of low status is likely to benefit far more from negative ties with high-status actors and suffer more from negative ties with low-status actors than will an actor of high-status. To test our ideas, we analyze the phenomenon of "diss songs" in hip-hop music. A diss song is a song in which a rapper makes derogatory comments about another rapper, constituting a negative tie. We analyze the effects of negative ties among 53 rappers over 20 points in time on audience reaction as measured by record sales. We find that negative ties with high-status actors enhance future sales for low-status actors. However, negative ties with lower-status actors have no effect on the future sales of both low- and high-status actors. Just as some researchers have reported both positive and negative consequences of social capital, our study demonstrates that negative ties can also have both positive and negative outcomes.     

The Statistical Evaluation of Social Network Dynamics

A class of statistical models is proposed for longitudinal network data. The dependent variable is the changing (or evolving) relation network, represented by two or more observations of a directed graph with a fixed set of actors. The network evolution is modeled as the consequence of the actors making new choices, or withdrawing existing choices, on the basis of functions, with fixed and random components, that the actors try to maximize. Individual and dyadic exogenous variables can be used as covariates. The change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and these covariates. The existing network structure is a dynamic constraint for the evolution of the structure itself. The models are continuous-time Markov chain models that can be implemented as simulation models. The model parameters are estimated from observed data. For estimating and testing these models, statistical procedures are proposed that are based on the method of moments. The statistical procedures are implemented using a stochastic approximation algorithm based on computer simulations of the network evolution process.     

Actor non-response in valued social networks: The impact of different non-response treatments on the stability of blockmodels

Social network data usually contain different types of errors. One of them is missing data due to actor non-response. This can seriously jeopardize the results of analyses if not appropriately treated. The impact of missing data may be more severe in valued networks where not only the presence of a tie is recorded, but also its magnitude or strength. Blockmodeling is a technique for delineating network structure. We focus on an indirect approach suitable for valued networks. Little is known about the sensitivity of valued networks to different types of measurement errors. As it is reasonable to expect that blockmodeling, with its positional outcomes, could be vulnerable to the presence of non-respondents, such errors require treatment. We examine the impacts of seven actor non-response treatments on the positions obtained when indirect blockmodeling is used. The start point for our simulation are networks whose structure is known. Three structures were considered: cohesive subgroups, core-periphery, and hierarchy. The results show that the number of non-respondents, the type of underlying blockmodel structure, and the employed treatment all have an impact on the determined partitions of actors in complex ways. Recommendations for best practices are provided.