Random sampling of alters from networks: A promising direction in egocentric network research

The social network perspective has great potential for advancing knowledge of social mechanisms in many fields. However, collecting egocentric (i.e., personal) network data is costly and places a heavy burden on respondents. This is especially true of the task used to elicit information on ties between network members (i.e., alter-alter ties or density matrix), which grows exponentially in length as network size increases. While most existing national surveys circumvent this problem by capping the number of network members that can be named, this strategy has major limitations. Here, we apply random sampling of network members to reduce cost, respondent burden, and error in network studies. We examine the effectiveness and reliability of random sampling in simulated and real-world egocentric network data. We find that in estimating sample/population means of network measures, randomly selecting a small number of network members produces only minor errors, regardless of true network size. For studies that use network measures in regressions, randomly selecting the mean number of network members (e.g., randomly selecting 10 alters when mean network size is 10) is enough to recover estimates of network measures that correlate close to 1 with those of the full sample. We conclude with recommendations for best practices that will make this versatile but resource intensive methodology accessible to a wider group of researchers without sacrificing data quality.     

Exponential Family Models for sampled and census Network data

Much progress has been made on the development of statistical methods for network analysis in the past ten years, building on the general class of exponential family random graph (ERG) network models first introduced by Holland and Leinhardt (1981) . Recent examples include models for Markov graphs, "p*" models, and actor‐oriented models. For empirical application, these ERG models take a logistic form, and require the equivalent of a network census: data on all dyads within the network. In a largely separate stream of research, conditional log‐linear (CLL) models have been adapted for analyzing locally sampled ("egocentric") network data. While the general relation between log‐linear and logistic models is well known and has been exploited in the case of a priori blockmodels for networks, the relation for the CLL models is different due to the treatment of absent ties. For fully saturated tie independence models, CLL and ERG are equivalent and related via Bayes' rule. For other tie independence models, the two do not yield equivalent predicted values, but we show that in practice the differences are unlikely to be large. The alternate conditioning in the two models sheds light on the relationship between local and complete network data, and the role that models can play in bridging the gap between them.     

Network autocorrelation models with egocentric data

Network autocorrelation models have been widely used for decades to model the joint distribution of the attributes of a network's actors. This class of models can estimate both the effect of individual characteristics as well as the network effect, or social influence, on some actor attribute of interest. Collecting data on the entire network, however, is very often infeasible or impossible if the network boundary is unknown or difficult to define. Obtaining egocentric network data overcomes these obstacles, but as of yet there has been no clear way to model this type of data and still appropriately capture the network effect on the actor attributes in a way that is compatible with a joint distribution on the full network data. This paper adapts the class of network autocorrelation models to handle egocentric data. The proposed methods thus incorporate the complex dependence structure of the data induced by the network rather than simply using ad hoc measures of the egos’ networks to model the mean structure, and can estimate the network effect on the actor attribute of interest. The vast quantities of unknown information about the network can be succinctly represented in such a way that only depends on the number of alters in the egocentric network data and not on the total number of actors in the network. Estimation is done within a Bayesian framework. A simulation study is performed to evaluate the estimation performance, and an egocentric data set is analyzed where the aim is to determine if there is a network effect on environmental mastery, an important aspect of psychological well-being.     

On the robustness of centrality measures under conditions of imperfect data

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

Are respondents more likely to list alters with certain characteristics?: Implications for name generator data

Analyses of egocentric networks make the implicit assumption that the list of alters elicited by name generators is a complete list or representative sample of relevant alters. Based on the literature on free recall tasks and the organization of people in memory, I hypothesize that respondents presented with a name generator are more likely to name alters with whom they share stronger ties, alters who are more connected within the network, and alters with whom they interact in more settings. I conduct a survey that presents respondents with the GSS name generator and then prompts them to remember other relevant alters whom they have not yet listed. By comparing the alters elicited before and after prompts I find support for the first two hypotheses. I then go on to compare network-level measures calculated with the alters elicited by the name generator to the same measures calculated with data from all alters. These measures are not well correlated. Furthermore, the degree of underestimation of network size is related to the networks’ mean closeness, density, and mean duration of relationships. Higher values on these variables result in more accurate estimation of network size. This suggests that measures of egocentric network properties based on data collected using a single name generator may have high levels of measurement error, possibly resulting in misestimation of how these network properties relate to other variables.     

Egocentric and sociocentric measures of network centrality

Egocentric centrality measures (for data on a node’s first-order zone) parallel to Freeman’s [Social Networks 1 (1979) 215] centrality measures for complete (sociocentric) network data are considered. Degree-based centrality is in principle identical for egocentric and sociocentric network data. A closeness measure is uninformative for egocentric data, since all geodesic distances from ego to other nodes in the first-order zone are 1 by definition. The extent to which egocentric and sociocentric versions of Freeman’s betweenness centrality measure correspond is explored empirically. Across seventeen diverse networks, that correspondence is found to be relatively close—though variations in egocentric network composition do lead to some notable differences in egocentric and sociocentric betweennness. The findings suggest that research design has a relatively modest impact on assessing the relative betweenness of nodes, and that a betweenness measure based on egocentric network data could be a reliable substitute for Freeman’s betweenness measure when it is not practical to collect complete network data. However, differences in the research methods used in sociocentric and egocentric studies could lead to additional differences in the respective betweenness centrality measures.     

Measurement error in network data: A re-classification

Research on measurement error in network data has typically focused on missing data. We embed missing data, which we term false negative nodes and edges, in a broader classification of error scenarios. This includes false positive nodes and edges and falsely aggregated and disaggregated nodes. We simulate these six measurement errors using an online social network and a publication citation network, reporting their effects on four node-level measures – degree centrality, clustering coefficient, network constraint, and eigenvector centrality. Our results suggest that in networks with more positively-skewed degree distributions and higher average clustering, these measures tend to be less resistant to most forms of measurement error. In addition, we argue that the sensitivity of a given measure to an error scenario depends on the idiosyncracies of the measure's calculation, thus revising the general claim from past research that the more ‘global’ a measure, the less resistant it is to measurement error. Finally, we anchor our discussion to commonly-used networks in past research that suffer from these different forms of measurement error and make recommendations for correction strategies.     

A comparison of email networks and off-line social networks: A study of a medium-sized bank

Recently there has been a surge in the availability of online data concerning the connections between people, and these online data are now widely used to map the social structure of communities. There has been little research, however, on how these new types of relational data correspond to classical measures of social networks. To fill this gap, we contrast the structure of an email network with the underlying friendship, communication, and advice seeking networks. Our study is an explorative case study of a bank, and our data contains emails among employees and a survey of the ego networks of the employees. Through calculating correlations with QAP standard errors and estimating exponential random graph (ERG) models, we find that although the email network is related to the survey-based social networks, email networks are also significantly different: while off-line social networks are strongly shaped by gender, tenure, and hierarchical boundaries, the role of these boundaries are much weaker in the email network.     

Connecting for success: Egocentric network types among underrepresented minority students at college

This study investigates the egocentric networks of underrepresented minority students. Adjustment to college for underrepresented minority students often means balancing between the origin (i.e., low-status family) and host (i.e., academic) environments, requiring students to develop different coping strategies. The analysis included 124 Roma university students. We employed the contact diary method and the EgoNet program for mapping students’ egocentric networks. We explored classification among egocentric networks by cluster analysis based on the network proportion of the origin, fellow, and host groups. Three clusters were distinguished. The Host-based cluster proved to be the most advantageous, while the Balancing cluster appeared to be the most disadvantageous in terms of network resources, well-being, and trust. Partners’ ethnicity and educational attainment showed a strong connection with cluster membership. Our results highlight that small fellow communities can successfully interconnect the origin and host groups in students’ egocentric networks.     

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.     

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.     

Omission and commission errors in network cognition and network estimation using ROC curve

Network studies on cognitive social structures collect relational data on respondents’ direct ties and their perception of ties among all other individuals in the network. When reporting their perception networks, respondents commit two types of errors, namely, omission (false negatives) and commission (false positives) errors. We first assess the relationship between these two error types, and their contributions on overall respondent accuracy. Next we propose a method for estimating networks based on perceptions of a random sample of respondents from a bounded social network, which utilizes the receiver operator characteristic curve for balancing the tradeoffs between omission and commission errors.     

Conditionally exponential random models for individual properties and network structures: Method and application

Exponential random models have been widely adopted as a general probabilistic framework for complex networks and recently extended to embrace broader statistical settings such as dynamic networks, valued networks or two-mode networks. Our aim is to provide a further step into the generalization of this class of models by considering sample spaces which involve both families of networks and nodal properties verifying combinatorial constraints. We propose a class of probabilistic models for the joint distribution of nodal properties (demographic and behavioral characteristics) and network structures (friendship and professional partnership). It results in a general and flexible modeling framework to account for homophily in social structures. We present a Bayesian estimation method based on the full characterization of their sample spaces by systems of linear constraints. This provides an exact simulation scheme to sample from the likelihood, based on linear programming techniques. After a detailed analysis of the proposed statistical methodology, we illustrate our approach with an empirical analysis of co-authorship of journal articles in the field of neuroscience between 2009 and 2013.     

Equicentrality and network centralization: A micro–macro linkage

This paper investigates a linkage between micro- and macrostructures as an intrinsic property of social networks. In particular, it examines the linkage between equicentrality [Kang, S.M., 2007. A note on measures of similarity based on centrality. Social Networks 29, 137–142] as a conceptualization of a microstructural process (i.e., the likelihood of social actors to be connected with similarly central others) and network centralization as a macrostructural construct, and shows that they have a negative linear association. In other words, when actors are connected with similarly central alters (i.e., high equicentrality), the overall network centralization is low. Conversely, when highly central actors are connected with low-centrality actors (i.e., low equicentrality), the overall network centralization is high. The relationship between degree equicentrality and degree centralization is more significant in observed networks, especially those evolving over time, as compared to random networks. An application of this property is given by venture capital co-investment networks.     

Networks and geography: Modelling community network structures as the outcome of both spatial and network processes

This paper focuses on how to extend the exponential random graph models to take into account the geographical embeddedness of individuals in modelling social networks. We develop a hierarchical set of nested models for spatially embedded social networks, in which, following Butts (2002), an interaction function between tie probability and Euclidean distance between nodes is introduced. The models are illustrated by an empirical example from a study of the role of social networks in understanding spatial clustering in unemployment in Australia. The analysis suggests that a spatial effect cannot solely explain the emergence of organised network structure and it is necessary to include both spatial and endogenous network effects in the model.     

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.     

Measuring personal networks with daily contacts: a single-item survey question and the contact diary

This paper examines two extreme approaches that are alternatives to measure egocentric networks with network generators. The single-item approach to measure daily contacts differentiates the individuals effectively, corresponds closely with complex network measures, and reveals well how individuals vary in both expressive and instrumental returns, as supported by 14 large-scale probability surveys from three Chinese societies over a decade. This paper also draws upon three sets of sophisticated contact diaries, which yielded rich data about the circumstance of each contact, the alter's characteristics and the ego–alter relationship. Along with the diary approach, which offers sophisticated data about contacts, ties and networks, the single-item survey approach is another extreme yet straightforward measure of daily contacts.     

Linked Ego Networks: Improving estimate reliability and validity with respondent-driven sampling

Respondent-driven sampling (RDS) is currently widely used for the study of HIV/AIDS-related high risk populations. However, recent studies have shown that traditional RDS methods are likely to generate large variances and may be severely biased since the assumptions behind RDS are seldom fully met in real life. To improve estimation in RDS studies, we propose a new method to generate estimates with ego network data, which is collected by asking respondents about the composition of their personal networks, such as “what proportion of your friends are married?”. By simulations on an extracted real-world social network of gay men as well as on artificial networks with varying structural properties, we show that the precision of estimates for population characteristics is greatly improved. The proposed estimator shows superior advantages over traditional RDS estimators, and most importantly, the method exhibits strong robustness to the recruitment preference of respondents and degree reporting error, which commonly happen in RDS practice and may generate large estimate biases and errors for traditional RDS estimators. The positive results henceforth encourage researchers to collect ego network data for variables of interests by RDS, for both hard-to-access populations and general populations when random sampling is not applicable.     

Corporate competition: A self-organized network

A substantial number of studies have extended the work on universal properties in physical systems to complex networks in social, biological, and technological systems. In this paper, we present a complex networks perspective on interfirm organizational networks by mapping, analyzing and modeling the spatial structure of a large interfirm competition network across a variety of sectors and industries within the United States. We propose two micro-dynamic models that are able to reproduce empirically observed characteristics of competition networks as a natural outcome of a minimal set of general mechanisms governing the formation of competition networks. Both models, which utilize different approaches yet apply common principles to network formation give comparable results. There is an asymmetry between companies that are considered competitors, and companies that consider others as their competitors. All companies only consider a small number of other companies as competitors; however, there are a few companies that are considered as competitors by many others. Geographically, the density of corporate headquarters strongly correlates with local population density, and the probability two firms are competitors declines with geographic distance. We construct these properties by growing a corporate network with competitive links using random incorporations modulated by population density and geographic distance. Our new analysis, methodology and empirical results are relevant to various phenomena of social and market behavior, and have implications to research fields such as economic geography, economic sociology, and regional economic development.     

An introduction to exponential random graph (p) models for social networks

This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement.