A Family of Affiliation Indices for Two-Mode Networks*

By Tutzauer, Frank | Journal of Social Structure, January 1, 2013 | Go to article overview

A Family of Affiliation Indices for Two-Mode Networks*


Tutzauer, Frank, Journal of Social Structure


(ProQuest: ... denotes formulae omitted.)

Introduction

Two-mode networks, in which the nodes of a network are partitioned into two groups, or modes, have received increasing attention in the social network literature (e.g., Latapy, Magnien and Del Vecchio 2008; Wang, Sharpe, Robins and Pattison 2009). The type of two-mode networks considered in this article are often called affiliation networks, and we think of one of the modes as a set of actors and the other mode as a set of events. Actors are affiliated with each other by virtue of the events they mutually attend. A classic example comes from Davis, Gardner and Gardner (1941) in which the actors are 18 southern women and the events are 14 social gatherings attended by varying numbers of the women. Even though one uses the terms actors and events, the objects of study need not be literal actors and events. For example, in the affiliation network of Ferrer i Cancho and Solé (2001), the ?actors? are words and the ?events? are the sentences in which they appear.

Affiliation networks have been used to study subjects as diverse as interlocking corporate directorates (Koenig, Gogel and Sonquist 1979), community organizations (Crowe 2007), director affiliation through both corporate and noncorporate organizations (Barnes and Burkett 2010), academic collaboration (Moody 2004), political communication (Chung and Park 2010; Park and Thelwall 2008), human language (Ferrer i Cancho and Solé 2001; Zhou and Heineken 2009), computer- mediated communication (Cho and Lee 2008), and the disciplinary structure of academic organizations (Barnett and Danowski 1992; Chung, Lee, Barnett and Kim 2009; Doerfel and Barnett 1999; Lee 2008).

Every affiliation network has a representation as a bipartite graph, where an edge is placed between an actor and an event only if the actor attended that event. Because certain edges are prohibited (namely edges between actors and edges between events), affiliation networks present certain problems (and opportunities) when one attempts to apply standard network measures (Borgatti and Everett 1997). This article develops a family of affiliation indices specifically for two-mode networks that are represented by bipartite graphs. I also derive the cumulative distribution function of one of the indices from the family and illustrate the measures on both hypothetical and empirical networks.

The structure of the paper is as follows: First, I introduce basic mathematical terminology. Then I define a family of affiliation indices and derive the sampling distribution of one of these indices. I then illustrate the ideas on an empirical network, and offer some concluding thoughts.

Mathematical Preliminaries

The networks in this article are loopless and undirected. Every such network has a mathematical representation as a graph G = (V, L), where V is a finite, nonempty set of vertices or nodes and L is an irreflexive, symmetric relation on V. The elements of L are called edges or links. If (v, w) Î L, then the nodes v and w are neighbors and the edge (v, w) is incident with the nodes. The degree of a node v, symbolized d(v), is the number of neighbors it has (with the parenthetical material dropped when the node is understood). In a two-mode network, the nodes can be partitioned into two groups meaningful to the investigator; for example, men and women or authors and their papers. Most often, it is assumed that every edge is incident with exactly one node from each group. In such a case, the representation is a bipartite graph. The graph G = (V, L) is bipartite if V can be partitioned into two nonempty, disjoint subsets A = {v1, v2, ... , vn} and E = {e1, e2, ... , em} such that every edge is incident with exactly one node from A and exactly one node from E. Two-mode networks with a bipartite representation are called affiliation networks, and we think of A as a set of actors and E as a set of events. All edges are incident with exactly one actor and exactly one event. …

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