Garry Robins

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.

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 neardegeneracy 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.

Small Worlds Among Interlocking Directors: Network Structure and Distance in Bipartite Graphs

We describe a methodology to examine bipartite relational data structures as exemplified in networks of corporate interlocking. These structures can be represented as bipartite graphs of directors and companies, but direct comparison of empirical datasets is often problematic because graphs have different numbers of nodes and different densities. We compare empirical bipartite graphs to simulated random graph distributions conditional on constraints implicit in the observed datasets. We examine bipartite graphs directly, rather than simply converting them to two 1-mode graphs, allowing investigation of bipartite statistics important to connection redundancy and bipartite connectivity. We introduce a new bipartite clustering coefficient that measures tendencies for localized bipartite cycles. This coefficient can be interpreted as an indicator of inter-company and inter-director closeness; but high levels of bipartite clustering have a cost for long range connectivity. We also investigate degree distributions, path lengths, and counts of localized subgraphs. Using this new approach, we compare global structural properties of US and Australian interlocking company directors. By comparing observed statistics against those from the simulations, we assess how the observed graphs are structured, and make comparisons between them relative to the simulated graph distributions. We conclude that the two networks share many similarities and some differences. Notably, both structures tend to be influenced by the clustering of directors on boards, more than by the accumulation of board seats by individual directors; that shared multiple board memberships (multiple interlocks) are an important feature of both infrastructures, detracting from global connectivity (but more so in the Australian case); and that company structural power may be relatively more diffuse in the US structure than in Australia.

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

Abstract 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.

Small and Other Worlds: Global Network Structures from Local Processes1

Using simulation, we contrast global network structures—in particular, small world properties—with the local patterning that generates the network. We show how to simulate Markov graph distributions based on assumptions about simple local social processes. We examine the resulting global structures against appropriate Bernoulli graph distributions and provide examples of stochastic global “worlds,” including small worlds, long path worlds, and nonclustered worlds with many four‐cycles. In light of these results we suggest a locally specified social process that produces small world properties. In examining movement from structure to randomness, parameter scaling produces a phase transition at a “temperature” where regular structures “melt” into stochastically based counterparts. We provide examples of “frozen” structures, including “caveman” graphs, bipartite structures, and cyclic structures.

Neighborhood–Based Models For Social Networks

We argue that social networks can be modeled as the outcome of processes that occur in overlapping local regions of the network, termed local social neighborhoods. Each neighborhood is conceived as a possible site of interaction and corresponds to a subset of possible network ties. In this paper, we discuss hypotheses about the form of these neighborhoods, and we present two new and theoretically plausible ways in which neighborhood-based models for networks can be constructed. In the first, we introduce the notion of a setting structure, a directly hypothesized (or observed) set of exogenous constraints on possible neighborhood forms. In the second, we propose higher-order neighborhoods that are generated, in part, by the outcome of interactive network processes themselves. Applications of both approaches to model construction are presented, and the developments are considered within a general conceptual framework of locale for social networks. We show how assumptions about neighborhoods can be cast within a hierarchy of increasingly complex models; these models represent a progressively greater capacity for network processes to “reach” across a network through long cycles or semipaths. We argue that this class of models holds new promise for the development of empirically plausible models for networks and network-based processes.

Network models for social selection processes

Abstract We present network models for social selection processes, based on the p∗ class of models. Social selection occurs when individuals form social relationships on the basis of certain characteristics they possess. Similarity is a common hypothesis for selection processes, but one that is usually framed dyadically. Structural balance approaches move beyond dyadic conceptualizations and require more sophisticated modeling. The two-block chain graph approach of p∗ social influence models is adapted to allow individual attribute variables to be predictors of network ties. Using a range of dependence assumptions, we present a hierarchy of increasingly complex selection models, including models for continuous attribute measures, which in their simplest form may be assumed to be linear. The models have scope, however, for more complex functional formulations so that more specific hypotheses may be investigated by postulating a particular functional form. Our empirical examples illustrate how dyadic selection may be transmuted into structural effects, and how the absence of dyadic selection may still mask a subtle higher order selection effect as individuals “position” themselves within a wider social environment. In conclusion, we discuss the links between social influence and social selection models.

Logit models and logistic regressions for social networks: III. Valued relations

This paper generalizes thep* model for dichotomous social network data (Wasserman & Pattison, 1996) to the polytomous case. The generalization is achieved by transforming valued social networks into three-way binary arrays. This data transformation requires a modification of the Hammersley-Clifford theorem that underpins thep* class of models. We demonstrate that, provided that certain (non-observed) data patterns are excluded from consideration, a suitable version of the theorem can be developed. We also show that the approach amounts to a model for multiple logits derived from a pseudo-likelihood function. Estimation within this model is analogous to the separate fitting of multinomial baseline logits, except that the Hammersley-Clifford theorem requires the equating of certain parameters across logits. The paper describes how to convert a valued network into a data array suitable for fitting the model and provides some illustrative empirical examples.

Obesity-related behaviors in adolescent friendship networks

This study examines obesity-related behaviors within adolescent friendship networks, because adolescent peers have been identified as being important determinants of many health behaviors. We applied ERGM selection models for single network observations to determine if close adolescent friends engage in similar behaviors and to explore associations between behavior and popularity. Same-sex friends were found to be similar on measures of organized physical activity in two out of three school-based friendship networks. Female friends were found to engage in similar screen-based behaviors, and male friends tended to be similar in their consumption of high-calorie foods. Popularity (receiving ties) was also associated with some behaviors, although these effects were gender specific and differed across networks.

Exponential random graph (p*) models for affiliation networks

Abstract Recent advances in Exponential Random Graph Models (ERGMs), or p ∗ models, include new specifications that give a much better chance of model convergence for large networks compared with the traditional Markov models. Simulation based MCMC maximum likelihood estimation techniques have been developed to replace the pseudolikelihood method. To date most work on ERGMs has focused on one-mode networks, with little done in the case of affiliation networks with two or more types of nodes. This paper proposes ERGMs for two-mode affiliation networks drawing on the recent advances for one-mode networks, including new two-mode specifications. We investigate features of the models by simulation, and compared the goodness of fit results obtained using the maximum likelihood and pseudolikelihood approaches. We introduce a new approach to goodness of fit for network models, using a heuristic based on Mahalanobis distance. The classic Southern Women data and Australian Interlocking Director data are used as examples to show that the ERGM with the newly specified statistics is a powerful tool for statistical analysis of affiliation networks.