Samuel Leinhardt

Stochastic blockmodels: First steps

Abstract A stochastic model is proposed for social networks in which the actors in a network are partitioned into subgroups called blocks. The model provides a stochastic generalization of the blockmodel. Estimation techniques are developed for the special case of a single relation social network, with blocks specified a priori. An extension of the model allows for tendencies toward reciprocation of ties beyond those explained by the partition. The extended model provides a one degree-of-freedom test of the model. A numerical example from the social network literature is used to illustrate the methods.

An Exponential Family of Probability Distributions for Directed Graphs

Abstract Directed graph (or digraph) data arise in many fields, especially in contemporary research on structures of social relationships. We describe an exponential family of distributions that can be used for analyzing such data. A substantive rationale for the general model is presented, and several special cases are discussed along with some possible substantive interpretations. A computational algorithm based on iterative scaling procedures for use in fitting data is described, as are the results of a pilot simulation study. An example using previously reported empirical data is worked out in detail. An extension to multiple relationship data is discussed briefly.

A Method for Detecting Structure in Sociometric Data

The authors focus on developing standardized measures for models of structure in interpersonal relations. A theorem is presented which yields expectations and variances for measures based on triads. Random models for these measures are discussed and the procedure is carried out for a model of a partial order. This model contains as special cases a number of previously suggested models, including the structural balance model of Cartwright and Harary, Daviss clustering model, and the ranked-clusters model of Davis and Leinhardt. In an illustrative exmaple, eight sociograms are analyzed and the general model is compared with the special case of ranked clusters.

Local Structure in Social Networks

This chapter is part of a continuing research series and reports work that is collaborative in every respect. The order of our names on this and our previous reports is alphabetical. National Science Foundation Grants GS-39778 to Carnegie-Mellon University and GJ-1 154X2 to the National Bureau of Economic Research, Inc., provided financial support. We are grateful to James A. Davis, J. Richard Dietrich, and Christopher Winship for aid in conducting this research and to Richard Hill for computer programing. This chapter was written when Paul Holland was with the Computer Research Center for Economics and Management Science of the National Bureau of Economic Research, Inc.

A dynamic model for social networks

A continuous‐time binary‐matrix‐valued Markov chain is used to model the process by which social structure effects individual behavior. The model is developed in the context of sociometric networks of interpersonal affect. By viewing the network as a time‐dependent stochastic process it is possible to construct transition intensity equations for the probability that choices between group members will change. These equations can contain parameters for structural effects. Empirical estimates of the parameters can be interpreted as measures of structural tendencies. Some elementary processes are described and the application of the model to cross‐sectional data is explained in terms of the steady state solution to the process.

The structural implications of measurement error in sociometry

Measurement error, an inherent quality of any empirical data collection technique, is discussed in the context of sociometric data. These data have long been assumed to possess face validity and to be the data of choice in any study of the sentiment structure of small scale social systems. However, it is argued that while methods of sociometric analysis have become increasingly more sophisticated they have failed to yield unequivocal results because they do not distinguish structural complexity from measurement error. Through a discussion of increasingly more complex examples the distortion laden character of most sociometric data is illustrated. This distortion is introduced by the formalities of the sociometric test and it will not be removed by developing increasingly more sophisticated structural models or throwing out some of the data. Instead, when issues concerning the nature of specific relational networks are raised data of much higher quality than those which are commonly available are required....

An Omnibus Test for Social Structure Using Triads

A general or omnibus test of structure in social network data is proposed. The test exploits all of the information contained in the triad census. Analogous to the classical F-test for contrasts among means, the proposed test involves finding a weighting vector which maximizes a test statistic, τ 2 (max), in the context ofan empirical data matrix and then determining whether this quantity is statistically significant by reference to a table of the chi-square distribution. An insignificant value of τ2 (max) implies that the structure of the network data matrix is ran dom, and, therefore, that the search for recognizable or substantively meaning ful pattern in the data may be subject to artifactual discoveries. Empirical results are presented which indicate that, of the networks commonly studied by social researchers, some have random structure, others have nonrandom structure and exhibit strong indications of transitivity and still others, with strong indications of nonrandom structure, do not exhibit strong indications of transitivity.

Developmental Change in the Sentiment Structure of Children's Groups

Current theories of cognitive development in children and those of structure in small-scale social systems when jointly considered suggest that the social organization of childrens peer groups will demonstrate developmental trends. This inference is tested by measuring transitive organization in 118 positive affect sociograms of childrens classroom groups and regressing these measurements on an age variable and other variables associated with the sociometric data. The analyses indicate that a statistically significant positive association exists between transitive organization and age and that this association is independent of variation in choices, made and received per group member, pair proportions, group size or sex composition.

Teaching Regressions an Exploratory Approach

Abstract An applied statistics and data-analysis course designed for students of public management and policy analysis, but suitable as an introductory graduate-level applied course in other contexts, is discussed. The course, Quantitative Methods for Public Management (QMPM), is a departure from traditional instruction in statistics. It uses subject-matter hierarchies to schedule the presentation of substantive material, and it integrates exploratory data analysis (EDA) and standard classical techniques. This integration is accomplished by using exploratory methods to clarify and evaluate analyses performed with classical procedures. The course, taught since 1975 at Carnegie-Mellon Universitys School of Urban and Public Affairs, has been evaluated experimentally through a randomized assignment of students to either a traditional introductory statistics course or QMPM. We concentrate here on the QMPM approach to teaching regression.

Characteristics of individuals who identify a regular source of medical care.

Having a source where medical services are regularly received is an antecedent to securing high quality medical care; it facilitates access and indicates that the individual is not alienated from the health care delivery system. In this paper we develop models to characterize individuals, both children and adults, who claim a regular source of care. The models are estimated using a logit analysis (since the dependent variable is 0-1) applied to survey data on residents of East Palo Alto, California. These data indicate that in this low-income, predominately black population the most important factor influencing whether a child will have a regular source of medical care is whether the parents have a regular source. For adults, the anticipated need for care (as measured by health status), time in community, and sex were all found to be important. The type of individual least likely to have a regular source of care is a low-income, unmarried male who is in good health and is a recent arrival to the community. The individuals most likely to need easy access to medical care and continuity of care are most likely to have a regular source of care, and vice versa.