VMASC Statistics and Social Network Analysis Project Report

Social network data are typically characterized by a set of binary link variables yi,j measured on pairs a set of objects or nodes. The presence of a link from i to j is expressed as yi,j = 1, and the absence of such a link by yi,j = 0. The situation in which it is not known whether or not there is a link from i to j is denoted as yi,j = NA, indicating the information about the relationship from i to j is “not available.” Some network data is undirected, in which yi,j = yj,i by design. Examples of undirected data include co-membership data, sexual networks and needle-sharing networks. Data in which it possible that yi,j 6= yj,i is called directed data, and include types of communication and other social interaction data. Social network data is a particular instance of so-called relational data, which describes generic pairwise information about relationships among a set of objects. The basic notation for relational data is the same as in social network data, except that the measurement yi,j is no longer required to be binary. Relational data include such things as export and import data among countries as well as length or frequency of communications among a set of actors. Data on the relationships among n actors or nodes can be expressed as an n×n matrix Y with an undefined diagonal, typically called a sociomatrix. Frequently accompanying such relational data are vectors of predictor variables xi,j , which may be specific to nodes or pairs. For example, among a set of schoolchildren, xi,j could contain demographic information about the pair i, j such as age, sex, SES and co-residence in the same neighborhood. Information on p predictor variables measured on each pair can be expressed with an n× n× p-dimensional array X. A common approach to the analysis of such data is to fit a statistical model relating Y to X. A statistical model can provide many forms of inference, such as a description of the association ∗Departments of Statistics, Biostatistics and the Center for Statistics and the Social Sciences, University of