Alter composition with overlapping group memberships

In this paper, we examine common methods of analyzing network data in which we characterize nodes via categorical attributes. For example, we consider alter composition (counting the number of alters of a given type), alter heterogeneity (measuring the diversity of ego's alters), homophily (similarity between ego and their alters) and Gould-Fernandez brokerage (assessing how often a node is in a position to broker between different categories). All of these techniques assume that all nodes belong to one and only one category -- i.e., the categorical attribute forms a node partition. However, there are a number of circumstances in which network actors belong to multiple categories. For example a manager might belong to multiple teams or project groups, while a teacher may teach more than one subject. We present a general approach for generalizing measures originally designed for mutually exclusive categories to the case where we have multiple memberships. Instead of a categorical vector, our method assumes a node by category membership matrix that is row-stochastic (e.g., the proportion of effort spent on each of several projects). Multiplying the adjacency matrix by this indicator matrix gives the indirect affinity each node has to each category through its alters. In the special case where each row has a single non-zero value (i.e., 1), the product counts the number of alters in each category, providing a true generalization of existing measures. In addition, we extend Burt's structural holes measures to not only take into account category membership, but also handle the case of multiple memberships.