Analyzing Data from Multivariate Directed Graphs: An Application to Social Networks

摘要

A multivariate directed graph consists of a set of g nodes, and a family of directed arcs (one for each relation) connecting pairs of nodes. Such multivariate directed graphs provide natural representations for social networks. In this paper, methods to analyse a network of 73 organizations in a Midwest American community linked by three types of relations is considered: information, money, and support. The resulting data set, described by Galaskiewicz and Marsden (1978), involves 3 x 73 x 72 = 15,768 possible arcs or 'observations'. The report describes a class of stochastic loglinear models for multivariate directed graphs, demonstrate how they can be fit to the data using generalized iterative scaling of Darroch and Ratcliff (1972), and explain the connection between these models and variants on standard loglinear models for multidimensional contingency tables discussed by Bishop, Fienberg, and Holland (1975). It also considers a disaggregation of the organizations into sub-groups, and demonstrate how to adapt the models to explore the intra- and inter-group relationships. These methods generalize research of Holland and Leinhardt (1980), who develop a model for dyadic relationships in univariate directed graph data. The paper includes a detailed analysis of the Galaskiewicz-Marsden data.