Impact of Survey Design on Estimation of Exponential-Family Random Graph Models from Egocentrically-Sampled Data.

Egocentric sampling of networks selects a subset of nodes ("egos") and collects information from them on themselves and their immediate network neighbours ("alters"), leaving the rest of the nodes in the network unobserved. This design is popular because it is relatively inexpensive to implement and can be integrated into standard sample surveys. Recent methodological developments now make it possible to statistically analyse this type of network data with Exponential-family Random Graph Models (ERGMs). This provides a framework for principled statistical inference, and the fitted models can in turn be used to simulate complete networks of arbitrary size that are consistent with the observed sample data, allowing one to infer the distribution of whole-network properties generated by the observed egocentric network statistics. In this paper, we discuss how design choices for egocentric network studies impact statistical estimation and inference for ERGMs. The design choices include both measurement strategies (for ego and alter attributes, and for ego-alter and alter-alter ties) and sampling strategies (for egos and alters). We discuss the importance of harmonising measurement specifications across egos and alters, and conduct simulation studies to demonstrate the impact of sampling design on statistical inference, specifically stratified sampling and degree censoring.

Exponential-Family Random Graph Models for Multi-Layer Networks.

Multi-layer networks arise when more than one type of relation is observed on a common set of actors. Modeling such networks within the exponential-family random graph (ERG) framework has been previously limited to special cases and, in particular, to dependence arising from just two layers. Extensions to ERGMs are introduced to address these limitations: Conway-Maxwell-Binomial distribution to model the marginal dependence among multiple layers; a "layer logic" language to translate familiar ERGM effects to substantively meaningful interactions of observed layers; and nondegenerate triadic and degree effects. The developments are demonstrated on two previously published datasets.

INFERENCE FOR SOCIAL NETWORK MODELS FROM EGOCENTRICALLY SAMPLED DATA, WITH APPLICATION TO UNDERSTANDING PERSISTENT RACIAL DISPARITIES IN HIV PREVALENCE IN THE US.

Egocentric network sampling observes the network of interest from the point of view of a set of sampled actors, who provide information about themselves and anonymized information on their network neighbors. In survey research, this is often the most practical, and sometimes the only, way to observe certain classes of networks, with the sexual networks that underlie HIV transmission being the archetypal case. Although methods exist for recovering some descriptive network features, there is no rigorous and practical statistical foundation for estimation and inference for network models from such data. We identify a subclass of exponential-family random graph models (ERGMs) amenable to being estimated from egocentrically sampled network data, and apply pseudo-maximum-likelihood estimation to do so and to rigorously quantify the uncertainty of the estimates. For ERGMs parametrized to be invariant to network size, we describe a computationally tractable approach to this problem. We use this methodology to help understand persistent racial disparities in HIV prevalence in the US. We also discuss some extensions, including how our framework may be applied to triadic effects when data about ties among the respondent's neighbors are also collected.

On the Question of Effective Sample Size in Network Modeling: An Asymptotic Inquiry.

The modeling and analysis of networks and network data has seen an explosion of interest in recent years and represents an exciting direction for potential growth in statistics. Despite the already substantial amount of work done in this area to date by researchers from various disciplines, however, there remain many questions of a decidedly foundational nature - natural analogues of standard questions already posed and addressed in more classical areas of statistics - that have yet to even be posed, much less addressed. Here we raise and consider one such question in connection with network modeling. Specifically, we ask, "Given an observed network, what is the sample size?" Using simple, illustrative examples from the class of exponential random graph models, we show that the answer to this question can very much depend on basic properties of the networks expected under the model, as the number of vertices nV in the network grows. In particular, adopting the (asymptotic) scaling of the variance of the maximum likelihood parameter estimates as a notion of effective sample size, say neff, we show that whether the networks are sparse or not under our model (i.e., having relatively few or many edges between vertices, respectively) is sufficient to yield an order of magnitude difference in neff, from O(nV ) to [Formula: see text]. We then explore some practical implications of this result, using both simulation and data on food-sharing from Lamalera, Indonesia.

An approximation method for improving dynamic network model fitting.

There has been a great deal of interest recently in the modeling and simulation of dynamic networks, i.e., networks that change over time. One promising model is the separable temporal exponential-family random graph model (ERGM) of Krivitsky and Handcock, which treats the formation and dissolution of ties in parallel at each time step as independent ERGMs. However, the computational cost of fitting these models can be substantial, particularly for large, sparse networks. Fitting cross-sectional models for observations of a network at a single point in time, while still a non-negligible computational burden, is much easier. This paper examines model fitting when the available data consist of independent measures of cross-sectional network structure and the duration of relationships under the assumption of stationarity. We introduce a simple approximation to the dynamic parameters for sparse networks with relationships of moderate or long duration and show that the approximation method works best in precisely those cases where parameter estimation is most likely to fail-networks with very little change at each time step. We consider a variety of cases: Bernoulli formation and dissolution of ties, independent-tie formation and Bernoulli dissolution, independent-tie formation and dissolution, and dependent-tie formation models.

A Separable Model for Dynamic Networks.

Models of dynamic networks - networks that evolve over time - have manifold applications. We develop a discrete-time generative model for social network evolution that inherits the richness and flexibility of the class of exponential-family random graph models. The model - a Separable Temporal ERGM (STERGM) - facilitates separable modeling of the tie duration distributions and the structural dynamics of tie formation. We develop likelihood-based inference for the model, and provide computational algorithms for maximum likelihood estimation. We illustrate the interpretability of the model in analyzing a longitudinal network of friendship ties within a school.

Exponential-family random graph models for valued networks.

Exponential-family random graph models (ERGMs) provide a principled and flexible way to model and simulate features common in social networks, such as propensities for homophily, mutuality, and friend-of-a-friend triad closure, through choice of model terms (sufficient statistics). However, those ERGMs modeling the more complex features have, to date, been limited to binary data: presence or absence of ties. Thus, analysis of valued networks, such as those where counts, measurements, or ranks are observed, has necessitated dichotomizing them, losing information and introducing biases. In this work, we generalize ERGMs to valued networks. Focusing on modeling counts, we formulate an ERGM for networks whose ties are counts and discuss issues that arise when moving beyond the binary case. We introduce model terms that generalize and model common social network features for such data and apply these methods to a network dataset whose values are counts of interactions.

Computational Statistical Methods for Social Network Models.

We review the broad range of recent statistical work in social network models, with emphasis on computational aspects of these methods. Particular focus is applied to exponential-family random graph models (ERGM) and latent variable models for data on complete networks observed at a single time point, though we also briefly review many methods for incompletely observed networks and networks observed at multiple time points. Although we mention far more modeling techniques than we can possibly cover in depth, we provide numerous citations to current literature. We illustrate several of the methods on a small, well-known network dataset, Sampson's monks, providing code where possible so that these analyses may be duplicated.

Adjusting for Network Size and Composition Effects in Exponential-Family Random Graph Models.

Exponential-family random graph models (ERGMs) provide a principled way to model and simulate features common in human social networks, such as propensities for homophily and friend-of-a-friend triad closure. We show that, without adjustment, ERGMs preserve density as network size increases. Density invariance is often not appropriate for social networks. We suggest a simple modification based on an offset which instead preserves the mean degree and accommodates changes in network composition asymptotically. We demonstrate that this approach allows ERGMs to be applied to the important situation of egocentrically sampled data. We analyze data from the National Health and Social Life Survey (NHSLS).

Representing Degree Distributions, Clustering, and Homophily in Social Networks With Latent Cluster Random Effects Models.

Social network data often involve transitivity, homophily on observed attributes, clustering, and heterogeneity of actor degrees. We propose a latent cluster random effects model to represent all of these features, and we describe a Bayesian estimation method for it. The model is applicable to both binary and non-binary network data. We illustrate the model using two real datasets. We also apply it to two simulated network datasets with the same, highly skewed, degree distribution, but very different network behavior: one unstructured and the other with transitivity and clustering. Models based on degree distributions, such as scale-free, preferential attachment and power-law models, cannot distinguish between these very different situations, but our model does.

Fitting Position Latent Cluster Models for Social Networks with latentnet.

latentnet is a package to fit and evaluate statistical latent position and cluster models for networks. Hoff, Raftery, and Handcock (2002) suggested an approach to modeling networks based on positing the existence of an latent space of characteristics of the actors. Relationships form as a function of distances between these characteristics as well as functions of observed dyadic level covariates. In latentnet social distances are represented in a Euclidean space. It also includes a variant of the extension of the latent position model to allow for clustering of the positions developed in Handcock, Raftery, and Tantrum (2007). The package implements Bayesian inference for the models based on an Markov chain Monte Carlo algorithm. It can also compute maximum likelihood estimates for the latent position model and a two-stage maximum likelihood method for the latent position cluster model. For latent position cluster models, the package provides a Bayesian way of assessing how many groups there are, and thus whether or not there is any clustering (since if the preferred number of groups is 1, there is little evidence for clustering). It also estimates which cluster each actor belongs to. These estimates are probabilistic, and provide the probability of each actor belonging to each cluster. It computes four types of point estimates for the coefficients and positions: maximum likelihood estimate, posterior mean, posterior mode and the estimator which minimizes Kullback-Leibler divergence from the posterior. You can assess the goodness-of-fit of the model via posterior predictive checks. It has a function to simulate networks from a latent position or latent position cluster model.