Mark S. Handcock

Latent Space Approaches to Social Network Analysis

Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors. We develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved “social space.” We make inference for the social space within maximum likelihood and Bayesian frameworks, and propose Markov chain Monte Carlo procedures for making inference on latent positions and the effects of observed covariates. We present analyses of three standard datasets from the social networks literature, and compare the method to an alternative stochastic blockmodeling approach. In addition to improving on model fit for these datasets, our method provides a visual and interpretable model-based spatial representation of social relationships and improves on existing methods by allowing the statistical uncertainty in the social space to be quantified and graphically represented.

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.

A Bayesian Analysis of Kriging

This article is concerned with predicting for Gaussian random fields in a way that appropriately deals with uncertainty in the covariance function. To this end, we analyze the best linear unbiased prediction procedure within a Bayesian framework. Particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality, both real and perceived, of the prediction. These ideas are implemented using topographical data from Davis.

Respondent-Driven Sampling: An Assessment of Current Methodology

Respondent-driven sampling (RDS) employs a variant of a link-tracing network sampling strategy to collect data from hard-to-reach populations. By tracing the links in the underlying social network, the process exploits the social structure to expand the sample and reduce its dependence on the initial (convenience) sample. The current estimators of population averages make strong assumptions in order to treat the data as a probability sample. We evaluate three critical sensitivities of the estimators: (1) to bias induced by the initial sample, (2) to uncontrollable features of respondent behavior, and (3) to the without-replacement structure of sampling. Our analysis indicates: (1) that the convenience sample of seeds can induce bias, and the number of sample waves typically used in RDS is likely insufficient for the type of nodal mixing required to obtain the reputed asymptotic unbiasedness; (2) that preferential referral behavior by respondents leads to bias; (3) that when a substantial fraction of the target population is sampled the current estimators can have substantial bias. This paper sounds a cautionary note for the users of RDS. While current RDS methodology is powerful and clever, the favorable statistical properties claimed for the current estimates are shown to be heavily dependent on often unrealistic assumptions. We recommend ways to improve the methodology.

Goodness of Fit of Social Network Models

We present a systematic examination of a real network data set using maximum likelihood estimation for exponential random graph models as well as new procedures to evaluate how well the models fit the observed networks. These procedures compare structural statistics of the observed network with the corresponding statistics on networks simulated from the fitted model. We apply this approach to the study of friendship relations among high school students from the National Longitudinal Study of Adolescent Health (AddHealth). We focus primarily on one particular network of 205 nodes, although we also demonstrate that this method may be applied to the largest network in the AddHealth study, with 2,209 nodes. We argue that several well-studied models in the networks literature do not fit these data well and demonstrate that the fit improves dramatically when the models include the recently developed geometrically weighted edgewise shared partner, geometrically weighted dyadic shared partner, and geometrically weighted degree network statistics. We conclude that these models capture aspects of the social structure of adolescent friendship relations not represented by previous models.

Inference in Curved Exponential Family Models for Networks

Network data arise in a wide variety of applications. Although descriptive statistics for networks abound in the literature, the science of fitting statistical models to complex network data is still in its infancy. The models considered in this article are based on exponential families; therefore, we refer to them as exponential random graph models (ERGMs). Although ERGMs are easy to postulate, maximum likelihood estimation of parameters in these models is very difficult. In this article, we first review the method of maximum likelihood estimation using Markov chain Monte Carlo in the context of fitting linear ERGMs. We then extend this methodology to the situation where the model comes from a curved exponential family. The curved exponential family methodology is applied to new specifications of ERGMs, proposed in an earlier article, having nonlinear parameters to represent structural properties of networks such as transitivity and heterogeneity of degrees. We review the difficult topic of implementing likelihood ratio tests for these models, then apply all these model-fitting and testing techniques to the estimation of linear and nonlinear parameters for a collaboration network between partners in a New England law firm.

RELATING RESOURCES TO A PROBABILISTIC MEASURE OF SPACE USE: FOREST FRAGMENTS AND STELLER'S JAYS

Many analytical techniques that assess resource selection focus on individual relocation points as the sample unit and classify resources as either used or available. Commonly, the relative use of each resource is quantified as the number of observations in each resource class or the proportional occurrence of a resource within a home range. We believe that a more accurate estimate can be summarized by a utilization distribution (UD). We present an analytical approach that explicitly incorporates a probabilistic measure of use, as defined by the UD. We used animal relocation points and fixed-kernel techniques to determine a UD within a home range. We related this probabilistic measure of use to categorical and continuous resource variables using multiple regression. Regression errors accounted for spatial autocorrelation so that the significance of regression coefficients could be appraised for each animal and averaged across animals. This allowed us to quantify the individualistic nature of resource sele...

AN APPROACH TO STATISTICAL SPATIAL-TEMPORAL MODELING OF METEOROLOGICAL FIELDS

Abstract In this article we develop a random field model for the mean temperature over the region in the northern United States covering eastern Montana through the Dakotas and northern Nebraska up to the Canadian border. The readings are temperatures at the stations in the U.S. historical climatological network. The stochastic structure is modeled by a stationary spatial-temporal Gaussian random field. For this region, we find little evidence of temporal dependence while the spatial structure is temporally stable. The approach strives to incorporate the uncertainty in estimating the covariance structure into the predictive distributions and the final inference. As an application of the model, we derive posterior distributions of the areal mean over time. A posterior distribution for the static areal mean is presented as a basis for calibrating temperature shifts by the historical record. For this region and season, the distribution indicates that under the scenario of a gradual increase of 5°F over 50 ye...

Relative distribution methods in the social sciences

and Motivation.- The Relative Distribution.- Location, Scale and Shape Decomposition.- Application: White Mens Earnings 1967-1997.- Summary Measures.- Application: Earnings by Race and Sex: 1967-1997.- Adjustment for Covariates.- Application: Comparing Wage Mobility in Two Eras.- Inference for the Relative Distribution.- Inference for Summary Measures.- The Relative Distribution for Discrete Data.- Application: Changes in the Distribution of Hours Worked.- Quantile Regression.