This paper identifies an important paradigm shift in social research on HIV transmission, drug abuse, and risk reduction research. The article describes the key research trends and the institutional support for social network analysis in the HIV and drug risk field for the past decade. Key hypotheses and recommended areas for future research are identified.

A flexible class of adaptive sampling designs is introduced for sampling in network and spatial settings. In the designs, selections are made sequentially with a mixture distribution based on an active set that changes as the sampling progresses, using network or spatial relationships as well as sample values. The new designs have certain advantages compared with previously existing adaptive and link-tracing designs, including control over sample sizes and of the proportion of effort allocated to adaptive selections. Efficient inference involves averaging over sample paths consistent with the minimal sufficient statistic. A Markov chain resampling method makes the inference computationally feasible. The designs are evaluated in network and spatial settings using two empirical populations: a hidden human population at high risk for HIV/AIDS and an unevenly distributed bird population.

Abstract. Respondent-driven sampling (RDS) is a recently introduced, and now widely used, technique for estimating disease prevalence in hidden populations. The sample is collected through a form of snowball sampling where current sample members recruit future sample members. In this paper we observe that respondent-driven sampling can be viewed as Markov chain Monte Carlo (MCMC) importance sampling. By establishing this connection, we are able to draw on the MCMC literature to address key RDS implementation and analysis issues. It was believed that the social network structure of the hidden population affected both the bias and variance of RDS estimates, but the precise nature of the relationship was unknown. Here we clarify that intuition by relating both to the second largest eigenvalue of the network transition matrix. In particular, we show segregation within the population effectively reduces sample size. We also show that sample size is effectively reduced by a sample design feature which allows sample members to recruit multiple future sample members. The paper concludes with suggestions for implementing and evaluating results from respondent-driven sampling studies. 1.

In classic statistical theory, if a random sample is drawn from a population whose underlying distribution is known, it may be assumed that the properties of the sample mirror those of the population (Snedecor and Cochran, 1972). On that cornerstone is built a statistical superstructure that permits estimation, hypothesis testing, assurance of internal validity, generalizability, and modeling. For a variety of actual sampling schemes — simple random, stratified, probability proportional to size, systematic, cluster, multistage — considerable mathematical work has established appropriate point estimate and variance formulas, and has defined the potential for bias and other threats to validity (Levy and Lemeshow, 1980). This body of work provides satisfying precision for the estimation of uncertainty in defining population characteristics. Random Graphs In the field of network analysis, sampling theory has been associated with defining the mathematical properties of random graphs. Though others preceded them, Erdos and Renyi (1959, 1960) are credited with establishing the theoretical base for estimation of such properties. During the past several decades considerable effort has been invested in describing graphs, and many familiar properties of social network have been established for random graphs. Investigators have explored the mean and variance of degree in a graph (Frank, 1980; Rapoport, 1979a); the probability that a graph will be connected (Gilbert, 1959); the distribution of connected components in a graph (Frank, 1978a; Ling, 1975; Naus and Rabinowitz, 1975); and general types of estimation in large graphs under various sampling schemes (Frank, 1980, 1981, 1978b) One specific type of network investigation — snowball sampling (Goodman, 1961) —

In this article we discuss a network sampling design that can be applied in drug abuse research at the community level. At this level often some partial sampling frame such as the register of a drug aid agency is available. This partial sampling frame can be used as the start of a network sample. Each selected registered drug abuser mentions his relationships with other drug abusers, and from those newly mentioned drug abusers who are not registered a second probability sample is drawn. Using this network sampling design the mean contact rates between clients, between clients and non registered drug abusers and between non registered drug abusers can be estimated despite the unknown total number of drug abusers. The design is illustrated by an analysis of the network data of the Heerlen Drug Monitoring System.