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 Finally, we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance.     

Advances in Exponential Random Graph (p*) Models Applied to a Large Social Network

Recent advances in statistical network analysis based on the family of exponential random graph (ERG) models have greatly improved our ability to conduct inference on dependence in large social networks (Snijders 2002, Pattison and Robins 2002, Handcock 2002, Handcock 2003, Snijders et al. 2006, Hunter et al. 2005, Goodreau et al. 2005, previous papers this issue). This paper applies advances in both model parameterizations and computational algorithms to an examination of the structure observed in an adolescent friendship network of 1,681 actors from the National Longitudinal Study of Adolescent Health (AddHealth). ERG models of social network structure are fit using the R package statnet, and their adequacy assessed through comparison of model predictions with the observed data for higher-order network statistics.For this friendship network, the commonly used model of Markov dependence leads to the problems of degeneracy discussed by Handcock (2002, 2003). On the other hand, model parameterizations introduced by Snijders et al (2006) and Hunter and Handcock (2006) avoid degeneracy and provide reasonable fit to the data. Degree-only models did a poor job of capturing observed network structure; those that did best included terms both for heterogeneous mixing on exogenous attributes (grade and self-reported race) as well as endogenous clustering. Networks simulated from this model were largely consistent with the observed network on multiple higher-order network statistics, including the number of triangles, the size of the largest component, the overall reachability, the distribution of geodesic distances, the degree distribution, and the shared partner distribution. The ability to fit such models to large datasets and to make inference about the underling processes generating the network represents a major advance in the field of statistical network analysis.     

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.     

A p* primer: logit models for social networks

A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others [Holland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33–65 (with discussion); Fienberg, S.E., Wasserman, S., 1981. Categorical data analysis of single sociometric relations. In: Leinhardt, S. (Ed.), Sociological Methodology 1981, San Francisco: Jossey-Bass, pp. 156–192; Fienberg, S.E., Meyer, M.M., Wasserman, S., 1985. Statistical analysis of multiple sociometric relations. Journal of the American Statistical Association, 80, pp. 51–67; Wasserman, S., Weaver, S., 1985. Statistical analysis of binary relational data: Parameter estimation. Journal of Mathematical Psychology. 29, pp. 406–427; Wasserman, S., 1987. Conformity of two sociometric relations. Psychometrika. 52, pp. 3–18] is the very strong independence assumption made on interacting individuals or units within a network or group. This limiting assumption is no longer necessary given recent developments on models for random graphs made by Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association. 81, pp. 832–842] and Strauss and Ikeda [Strauss, D., Ikeda, M., 1990. Pseudolikelihood estimation for social networks. Journal of the American Statistical Association. 85, pp. 204–212]. The resulting models are extremely flexible and easy to fit to data. Although Wasserman and Pattison [Wasserman, S., Pattison, P., 1996. Logit models and logistic regressions for social networks: I. An introduction to Markov random graphs and p*. Psychometrika. 60, pp. 401–426] present a derivation and extension of these models, this paper is a primer on how to use these important breakthroughs to model the relationships between actors (individuals, units) within a single network and provides an extension of the models to multiple networks. The models for multiple networks permit researchers to study how groups are similar and/or how they are different. The models for single and multiple networks and the modeling process are illustrated using friendship data from elementary school children from a study by Parker and Asher [Parker, J.G., Asher, S.R., 1993. Friendship and friendship quality in middle childhood: Links with peer group acceptance and feelings of loneliness and social dissatisfaction. Developmental Psychology. 29, pp. 611–621].     

Curved Exponential Family Models for Social Networks

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, and alternating k-twopath statistics of Snijders et al (2006) may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.     

Exponential random graph model specifications for bipartite networks—A dependence hierarchy

In this paper, we review the development of dependence structures for exponential random graph models for bipartite networks, and propose a hierarchy of dependence structures within which different dependence assumptions may be located. Based on this hierarchy, we propose a new set of model specifications by including bipartite graph configurations involving more than four nodes. We discuss the theoretical significance of the various effects that the extended models afford, and illustrate application of this hierarchy of models to several bipartite networks related to the political mobilization in Brazil in the early 1990s (Mische, 2007).     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time is the class of exponential random graph models (ERGMs), also known as p * models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, Markov chain Monte Carlo (MCMC) algorithms have been developed that produce approximate maximum likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data.
This paper proposes new specifications of exponential random graph models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistics are proposed: geometrically weighted degree distributions, alternating k -triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.     

A comparison of various approaches to the exponential random graph model: A reanalysis of 102 student networks in school classes

This paper describes an empirical comparison of four specifications of the exponential family of random graph models (ERGM), distinguished by model specification (dyadic independence, Markov, partial conditional dependence) and, for the Markov model, by estimation method (Maximum Pseudolikelihood, Maximum Likelihood). This was done by reanalyzing 102 student networks in 57 junior high school classes. At the level of all classes combined, earlier substantive conclusions were supported by all specifications. However, the different specifications led to different conclusions for individual classes. PL produced unreliable estimates (when ML is regarded as the standard) and had more convergence problems than ML. Furthermore, the estimates of covariate effects were affected considerably by controlling for network structure, although the precise specification of the structural part (Markov or partial conditional dependence) mattered less.     

A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models

The statistical modeling of social network data is difficult due to the complex dependence structure of the tie variables. Statistical exponential families of distributions provide a flexible way to model such dependence. They enable the statistical characteristics of the network to be encapsulated within an exponential family random graph (ERG) model. For a long time, however, likelihood-based estimation was only feasible for ERG models assuming dyad independence. For more realistic and complex models inference has been based on the pseudo-likelihood. Recent advances in computational methods have made likelihood-based inference practical, and comparison of the different estimators possible.     

Analyzing event stream dynamics in two-mode networks: An exploratory analysis of private communication in a question and answer community

Information about social networks can often be collected as event stream data. However, most methods in social network analysis are defined for static network snapshots or for panel data. We propose an actor oriented Markov process framework to analyze the structural dynamics in event streams. Estimated parameters are similar to what is known from exponential random graph models or stochastic actor oriented models as implemented in SIENA. We apply the methodology on a question and answer web community and show how the relevance of different kinds of one- and two-mode network structures can be tested using a new software.     

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.     

Exponential random graph models for multilevel networks

Modern multilevel analysis, whereby outcomes of individuals within groups take into account group membership, has been accompanied by impressive theoretical development (e.g. Kozlowski and Klein, 2000) and sophisticated methodology (e.g. Snijders and Bosker, 2012). But typically the approach assumes that links between groups are non-existent, and interdependence among the individuals derives solely from common group membership. It is not plausible that such groups have no internal structure nor they have no links between each other. Networks provide a more complex representation of interdependence. Drawing on a small but crucial body of existing work, we present a general formulation of a multilevel network structure. We extend exponential random graph models (ERGMs) to multilevel networks, and investigate the properties of the proposed models using simulations which show that even very simple meso effects can create structure at one or both levels. We use an empirical example of a collaboration network about French cancer research elites and their affiliations (0125 and 0120) to demonstrate that a full understanding of the network structure requires the cross-level parameters. We see these as the first steps in a full elaboration for general multilevel network analysis using ERGMs.     

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.     

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.     

Bayesian model selection for exponential random graph models

Exponential random graph models are a class of widely used exponential family models for social networks. The topological structure of an observed network is modelled by the relative prevalence of a set of local sub-graph configurations termed network statistics. One of the key tasks in the application of these models is which network statistics to include in the model. This can be thought of as statistical model selection problem. This is a very challenging problem—the posterior distribution for each model is often termed “doubly intractable” since computation of the likelihood is rarely available, but also, the evidence of the posterior is, as usual, intractable. The contribution of this paper is the development of a fully Bayesian model selection method based on a reversible jump Markov chain Monte Carlo algorithm extension of Caimo and Friel (2011) which estimates the posterior probability for each competing model.     

Stochastic weighted graphs: Flexible model specification and simulation

In most domains of network analysis researchers consider networks that arise in nature with weighted edges. Such networks are routinely dichotomized in the interest of using available methods for statistical inference with networks. The generalized exponential random graph model (GERGM) is a recently proposed method used to simulate and model the edges of a weighted graph. The GERGM specifies a joint distribution for an exponential family of graphs with continuous-valued edge weights. However, current estimation algorithms for the GERGM only allow inference on a restricted family of model specifications. To address this issue, we develop a Metropolis–Hastings method that can be used to estimate any GERGM specification, thereby significantly extending the family of weighted graphs that can be modeled with the GERGM. We show that new flexible model specifications are capable of avoiding likelihood degeneracy and efficiently capturing network structure in applications where such models were not previously available. We demonstrate the utility of this new class of GERGMs through application to two real network data sets, and we further assess the effectiveness of our proposed methodology by simulating non-degenerate model specifications from the well-studied two-stars model. A working R version of the GERGM code is available in the supplement and is incorporated in the GERGM CRAN package.     

Social context,spatial structure and social network structure

Frequently, social networks are studied in their own right with analyses devoid of contextual details. Yet contextual features – both social and spatial – can have impacts on the networks formed within them. This idea is explored with five empirical networks representing different contexts and the use of distinct modeling strategies. These strategies include network visualizations, QAP regression, exponential random graph models, blockmodeling and a combination of blockmodels with exponential random graph models within a single framework. We start with two empirical examples of networks inside organizations. The familiar Bank Wiring Room data show that the social organization (social context) and spatial arrangement of the room help account for the social relations formed there. The second example comes from a police academy where two designed arrangements, one social and one spatial, powerfully determine the relational social structures formed by recruits. The next example is an inter-organizational network that emerged as part of a response to a natural disaster where features of the improvised context helped account for the relations that formed between organizations participating in the search and rescue mission. We then consider an anthropological example of signed relations among sub-tribes in the New Guinea highlands where the physical geography is fixed. This is followed by a trading network off the Dalmatian coast where geography and physical conditions matter. Through these examples, we show that context matters by shaping the structure of networks that form and that a variety of network analytic tools can be mobilized to reveal how networks are shaped, in part, by social and spatial contexts. Implications for studying social networks are suggested.     

What makes a network go round? Exploring the structure of a strong component with exponential random graph models

A strong component is a subgraph in a directed network where, following the direction of ties, all nodes in the graph are reachable from one another. Mutual reachability implies that every node in the graph is theoretically able to send materials to and/or influence every other node suggesting that strong components are amongst the more egalitarian network structures. Despite this intriguing feature, they remain understudied. Using exponential random graph models (ERGM) for directed networks, we investigate the social and structural processes underlying the generation of strong components. We illustrate our argument using a network of 301 nodes and 703 personal lending ties from Renaissance Florence. ERGM shows that our strong component arises from triadic clustering alongside an absence of higher-order star structures. We contend that these processes produce a strong component with a hierarchical, rather than an egalitarian structure: while some nodes are deeply embedded in a dense network of exchange, the involvement of others is more tenuous. More generally, we argue that such tiered core-periphery strong components will predominate in networks where the social context creates conditions for an absence of preferential attachment alongside the presence of localized closure. Although disparate social processes can give rise to hierarchical strong components linked to these two structural mechanisms, in Florence they are associated with the presence of multiple dimensions of social status and the connectedness of participants across disparate network domains.     

Modelling a disease-relevant contact network of people who inject drugs

This study uses social network analysis to model a contact network of people who inject drugs (PWID) relevant for investigating the spread of an infectious disease (hepatitis C). Using snowball sample data, parameters for an exponential random graph model (ERGM) including social circuit dependence and four attributes (location, age, injecting frequency, gender) are estimated using a conditional estimation approach that respects the structure of snowball sample designs. Those network parameter estimates are then used to create a novel, model-dependent estimate of network size. Simulated PWID contact networks are created and compared with Bernoulli graphs. Location, age and injecting frequency are shown to be statistically significant attribute parameters in the ERGM. Simulated ERGM networks are shown to fit the collected data very well across a number of metrics. In comparison with Bernoulli graphs, simulated networks are shown to have longer paths and more clustering. Results from this study make possible simulation of realistic networks for investigating treatment and intervention strategies for reducing hepatitis C prevalence.     

Who benefits from network intervention programs? TERGM analysis across ten Philippine low-income communities

Building social capital and strengthening social networks among members of low-income communities has been recommended as a potential pathway out of poverty. However, it is not clear how network-strengthening interventions and community-based programs interact with pre-existing networks and power structures. We examine the impact of one such intervention in ten low income communities in the Philippines. The intervention is a standardized program of a faith-based organization implemented in thousands of communities in multiple countries. It brings together low-income individuals in each community for 16 weekly sessions about health, income generation, and Christian values. An important but yet unmeasured goal of the intervention is the strengthening of social networks among the participants. We measured the social networks before and after the intervention and analysed their changes both separately and jointly for all ten communities with temporal exponential random graph models (TERGM). We modelled the post-intervention network structures conditioning on the pre-intervention networks, pre-intervention node attributes, and attribute changes through the intervention. We found social engagement (measured by social visits to others) to moderate most consistently the effects of the intervention across the ten communities. Those who were more socially engaged consistently strengthened their networks through the intervention. By contrast, some network mechanisms strongly diverged between the communities. In particular, religiosity was positively associated with gaining social links through this faith-based intervention in some communities and negatively in others. Similar communities may in some aspects react to the same intervention in opposite ways—a phenomenon that should be further explored through studies of larger numbers of comparable networks.