Closure,connectivity and degree distributions: Exponential random graph (p*) models for directed social networks

The new higher order specifications for exponential random graph models introduced by Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology 36, 99–153] exhibit substantial improvements in model fit compared with the commonly used Markov random graph models. Snijders et al., however, concentrated on non-directed graphs, with only limited extensions to directed graphs. In particular, they presented a transitive closure parameter based on path shortening. In this paper, we explain the theoretical and empirical advantages in generalizing to additional closure effects. We propose three new triadic-based parameters to represent different versions of triadic closure: cyclic effects; transitivity based on shared choices of partners; and transitivity based on shared popularity. We interpret the last two effects as forms of structural homophily, where ties emerge because nodes share a form of localized structural equivalence. We show that, for some datasets, the path shortening parameter is insufficient for practical modeling, whereas the structural homophily parameters can produce useful models with distinctive interpretations. We also introduce corresponding lower order effects for multiple two-path connectivity. We show by example that the in- and out-degree distributions may be better modeled when star-based parameters are supplemented with parameters for the number of isolated nodes, sources (nodes with zero in-degrees) and sinks (nodes with zero out-degrees). Inclusion of a Markov mixed star parameter may also help model the correlation between in- and out-degrees. We select some 50 graph features to be investigated in goodness of fit diagnostics, covering a variety of important network properties including density, reciprocity, geodesic distributions, degree distributions, and various forms of closure. As empirical illustrations, we develop models for two sets of organizational network data: a trust network within a training group, and a work difficulty network within a government instrumentality.     

The consumer litigation industry: Chasing dragon kings in lawyer–client networks

The community of lawyers and their clients form a scale-free bipartite network that develops naturally as the outcome of the recommendation process through which lawyers form their client base. This process is an example of preferential attachment where lawyers with more clients are more likely to be recommended to new clients. Consumer litigation is an important market for lawyers. In large consumer societies, there always a significant amount of consumption disputes that escalate to court. In this paper we analyze a dataset of thousands of lawsuits, reconstructing the lawyer–client network embedded in the data. Analyzing the degree distribution of this network we noticed that it follows that of a scale-free network built by preferential attachment, but for a few lawyers with much larger client base than could be expected by preferential attachment. Incidentally, most of these also figured on a list put together by the judiciary of lawyers which openly advertised the benefits of consumer litigation. According to the code of ethics of their profession, lawyers should not stimulate clients into litigation, but it is not strictly illegal. From a network formation point of view, this stimulation can be seen as a separate growth mechanism than preferential attachment alone. In this paper we find that this composite growth can be detected by a simple statistical test, as simulations show that lawyers which use both mechanisms quickly become the “dragon-kings” of the distribution of the number of clients per lawyer.     

A latent space model for cognitive social structures data

This paper introduces a novel approach for modeling a set of directed, binary networks in the context of cognitive social structures (CSSs) data. We adopt a relativist approach in which no assumption is made about the existence of an underlying true network. More specifically, we rely on a generalized linear model that incorporates a bilinear structure to model transitivity effects within networks, and a hierarchical specification on the bilinear effects to borrow information across networks. This is a spatial model, in which the perception of each individual about the strength of the relationships can be explained by the perceived position of the actors (themselves and others) on a latent social space. A key goal of the model is to provide a mechanism to formally assess the agreement between each actors’ perception of their own social roles with that of the rest of the group. Our experiments with both real and simulated data show that the capabilities of our model are comparable with or, even superior to, other models for CSS data reported in the literature.     

If you do not know who knows what: Advice seeking under changing conditions of uncertainty after an acquisition

In this study we develop a model to explain the dynamics of advice seeking after an acquisition. We build on a theory of advice seeking that draws from prospect theory and expectancy theory. We theorize that immediately after an acquisition there is uncertainty about who knows what, but over time individuals become more aware of the expertise within the organization and they change their advice networks based upon this increased awareness. Our model examines four micro-processes of advice seeking: reciprocity, preferential attachment, transitivity, and legacy-firm tie preferences. To test our hypotheses we use post-acquisition data over four time periods in a recruitment consulting firm. Our longitudinal analysis uses a stochastic actor-orientated model and our results indicate that immediately after the acquisition individuals have a tendency to seek advice based upon reciprocity and preferential attachment. However, over time these tendencies diminish. Surprisingly, transitivity does not play a significant role, which suggests that other micro-processes such as reciprocity are dominant. In addition, individuals in the acquired firm have a tendency to make more ties and there is a preference for same firm ties in both legacy firms, with the tendency being higher in the acquired firm. Our findings add to theories on the process of advice seeking under conditions of uncertainty, on knowledge transfer processes in mergers and acquisitions, and the knowledge based view of the firm.     

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.     

Recent developments in exponential random graph (p*) models for social networks

This article reviews new specifications for exponential random graph models proposed by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology] and demonstrates their improvement over homogeneous Markov random graph models in fitting empirical network data. Not only do the new specifications show improvements in goodness of fit for various data sets, but they also help to avoid the problem of near-degeneracy that often afflicts the fitting of Markov random graph models in practice, particularly to network data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows estimation of parameters of exponential graph models for many (but not all) cases where it is impossible to estimate parameters of homogeneous Markov graph models. The new specifications were used to model a large number of classical small-scale network data sets and showed a dramatically better performance than Markov graph models. We also review three current programs for obtaining maximum likelihood estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with less accurate pseudo-likelihood estimates. Finally, we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance.     

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.     

Latent Class Factor and Cluster Models, Bi-Plots, and Related Graphical Displays

We propose an alternative method of conducting exploratory latent class analysis that utilizes latent class factor models, and compare it to the more traditional approach based on latent class cluster models. We show that when formulated in terms of R mutually independent, dichotomous latent factors, the LC factor model has the same number of distinct parameters as an LC cluster model with R+1 clusters. Analyses over several data sets suggest that LC factor models typically fit data better and provide results that are easier to interpret than the corresponding LC cluster models. We also introduce a new graphical "bi-plot" display for LC factor models and compare it to similar plots used in correspondence analysis and to a barycentric coordinate display for LC cluster models. New results on identification of LC models are also presented. We conclude by describing various model extensions and an approach for eliminating boundary solutions in identified and unidentified LC models, which we have implemented in a new computer program.     

Reciprocity,transitivity, and the mysterious three-cycle

Reciprocity and transitivity are the two most important structural mechanisms underlying friendship network evolution. While on their own they are understood in great detail, the relation between them is rarely studied systematically. Are friendships outside of social groups more or less likely to be reciprocated than friendships embedded in a group? Using a theoretical framework that focusses on the situations in which friends interact and the social structures that stabilise one-sided friendships, I propose that the tendency towards reciprocation of friendships within transitive groups is usually lower than outside of transitive groups. In a meta-analysis of two datasets including 29 friendship networks using stochastic actor-oriented models (SAOMs), the interaction between reciprocity and transitivity is analysed. Supporting the theoretical reasoning, the interaction is consistently negative. Second, the tendency against forming three-cycles in friendship networks, which was consistently found in previous studies, is shown to be spurious and a result of neglecting to control for the tendency against reciprocation in transitive groups. The tendency against three-cycles is commonly seen as an indicator that unreciprocated friendships indicate local hierarchy differences between individuals; this proposition has to be re-evaluated in light of the findings of this study. Future studies that analyse the evolution of friendship networks should consider modelling reciprocation in transitive triplets and potentially omit modelling three-cycles.     

Testing transitivity in graphs

The situation is considered whether a graph can be assumed to have been generated by a random model capturing more transitivity than a simple uniform model. Three different test quantities based on induced triad counts and local densities are used. A simulation study is made in order to estimate critical values of the tests for different significance levels. The powers of the tests are estimated against the Bernoulli triangle model, a simple random graph model in which the clustering and transitivity is higher than in the uniform model. The test based on the proportion of transitive triads has the highest power in most cases, but the test based on density difference (the difference between mean local density and overall graph density) is more powerful against models with high transitivity. The tests are applied to a large set of school class sociograms. In this situation, uniform randomness is rejected in favor of transitivity most frequently when the test based on the proportion of transitive triads out of the non-vacuously transitive triads is used. It is concluded that this test, which also performed reasonably well when applied to random data, is the best at detecting transitivity. Although the Bernoulli triangle model fits to the empirical data set better than the uniform model, there are fewer truly intransitive triads in the data than could be expected under either of the models.     

Ego centered social network and relationship quality: linking attachment security and relational models to network structure

According to Dunbar’s social brain hypothesis, brain capacity puts a limit on the size of social network in humans. At the same time, emotional-cognitive systems such as attachment and different relational models (Fiske, 1991) may also act as driving forces behind network structure underlying relationship quality of the various ego-centered layers. The aim of our study was to examine how these cognitive systems may contribute to network organization.We examined the two inner layers of the ego-network in 274 volunteers, the support and sympathy groups, as well as close family members in comparison to friends within the support group, in terms of attachment security and underlying relational models.     

Covariate-defined latent space random effects model

Latent factor models are a useful and intuitive class of models; one limitation is their inability to predict links in a dynamic network. We propose a latent space random effects model with a covariate-defined social space, where the social space is a linear combination of the covariates as estimated by an MCMC algorithm. The model allows for the prediction of links in a network; it also provides an interpretable framework to explain why people connect. We fit the model using the Adolescent Health Network dataset and three simulated networks to illustrate its effectiveness in recognizing patterns in the data.     

Does proximity matter? Distance dependence of adolescent friendships

Geographic proximity is a determinant factor of friendship. Friendship datasets that include detailed geographic information are scarce, and when this information is available, the dependence of friendship on distance is often modelled by pre-specified parametric functions or derived from theory without further empirical assessment. This paper aims to give a detailed representation of the association between distance and the likelihood of friendship existence and friendship dynamics, and how this is modified by a few basic social and individual factors. The data employed is a three-wave network of 336 adolescents living in a small Swedish town, for whom information has been collected on their household locations. The analysis is a three-step process that combines (1) nonparametric logistic regressions to unravel the overall functional form of the dependence of friendship on distance, without assuming it has a particular strength or shape; (2) parametric logistic regressions to construct suitable transformations of distance that can be employed in (3) stochastic models for longitudinal network data, to assess how distance, individual covariates, and network structure shape adolescent friendship dynamics. It was found that the log-odds of friendship existence and friendship dynamics decrease smoothly with the logarithm of distance. For adolescents in different schools the dependence is linear, and stronger than for adolescents in the same school. Living nearby accounts, in this dataset, for an aspect of friendship dynamics that is not explicitly modelled by network structure or by individual covariates. In particular, the estimated distance effect is not correlated with reciprocity or transitivity effects.     

Estimating network structure via random sampling: Cognitive social structures and the adaptive threshold method

This paper introduces and tests a novel methodology for measuring networks. Rather than collecting data to observe a network or several networks in full, which is typically costly or impossible, we randomly sample a portion of individuals in the network and estimate the network based on the sampled individuals’ perceptions on all possible ties. We find the methodology produces accurate estimates of social structure and network level indices in five different datasets. In order to illustrate the performance of our approach we compare its results with the traditional roster and ego network methods of data collection. Across all five datasets, our methodology outperforms these standard social network data collection methods. We offer ideas on applications of our methodology, and find it especially promising in cross-network settings.     

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).     

Missing data in networks: exponential random graph (p1) models for networks with non-respondents

Survey studies of complete social networks often involve non-respondents, whereby certain people within the “boundary” of a network do not complete a sociometric questionnaire—either by their own choice or by the design of the study—yet are still nominated by other respondents as network partners. We develop exponential random graph (p¹) models for network data with non-respondents. We model respondents and non-respondents as two different types of nodes, distinguishing ties between respondents from ties that link respondents to non-respondents. Moreover, if we assume that the non-respondents are missing at random, we invoke homogeneity across certain network configurations to infer effects as applicable to the entire set of network actors. Using an example from a well-known network dataset, we show that treating a sizeable proportion of nodes as non-respondents may still result in estimates, and inferences about structural effects, consistent with those for the entire network.If, on the other hand, the principal research focus is on the respondent-only structure, with non-respondents clearly not missing at random, we incorporate the information about ties to non-respondents as exogenous. We illustrate this model with an example of a network within and between organizational departments. Because in this second class of models the number of non-respondents may be large, values of parameter estimates may not be directly comparable to those for models that exclude non-respondents. In the context of discussing recent technical developments in exponential random graph models, we present a heuristic method based on pseudo-likelihood estimation to infer whether certain structural effects may contribute substantially to the predictive capacity of a model, thereby enabling comparisons of important effects between models with differently sized node sets.     

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.     

Relationships between landscape pattern metrics,vertical structure and surface urban Heat Island formation in a Colorado suburb

Change in land surface temperature (LST) due to urbanization is a critical influence on the ecohydrological function and human health in cities. Past research has emphasized land cover composition and planimetric landscape pattern as drivers of LST patterns, but there have been few empirical studies examining the effects of vertical structure on thermal variation in cities. We used agglomerative cluster analysis to group 5 ha grid cells with similar land cover composition from high-resolution land cover data for our Aurora, Colorado, USA study area. We then compared the importance of different planimetric landscape pattern metrics and lidar-derived vertical structure variables as predictors of LST estimated from Landsat 5 thermal band data for each group. Variation in LST between cluster groups was analyzed with landscape and vertical structure data using random forest regression models and conditional variable importance metrics. Cluster analysis produced 7 distinct groups differing in land cover composition, planimetric landscape pattern, and vertical structure. Clusters with greater tree cover and higher mean tree height were generally cooler, while the height difference between buildings and trees had high conditional variable importance in random forest regression models. We found that the specific planimetric landscape pattern and vertical structure metrics most important for individual clusters differed, suggesting that the relative importance of variables depends on land-use and land cover. Our results highlight the importance of including vertical structure in empirical analyses of urban LST.     

Random-Effects Modeling of Categorical Response Data

In many applications observations have some type of clustering, with observations within clusters tending to be correlated. A common instance of this occurs when each subject in the sample undergoes repeated measurement, in which case a cluster consists of the set of observations for the subject. One approach to modeling clustered data introduces cluster-level random effects into the model. The use of random effects in linear models for normal responses is well established. By contrast, random effects have only recently seen much use in models for categorical data. This chapter surveys a variety of potential social science applications of random effects modeling of categorical data. Applications discussed include repeated measurement for binary or ordinal responses, shrinkage to improve multiparameter estimation of a set of proportions or rates, multivariate latent variable modeling, hierarchically structured modeling, and cluster sampling. The models discussed belong to the class of generalized linear mixed models (GLMMs), an extension of ordinary linear models that permits nonnormal response variables and both fixed and random effects in the predictor term. The models are GLMMs for either binomial or Poisson response variables, although we also present extensions to multicategory (nominal or ordinal) responses. We also summarize some of the technical issues of model-fitting that complicate the fitting of GLMMs even with existing software.