# Notes: Tutzauer, F. (2013). A Family of Affiliation Indices for Two-Mode Networks

### 2017-06-15

## References

**Citekey**: @Tutzauer2013-oq

Tutzauer, F. (2013). A Family of Affiliation Indices for Two-Mode Networks. Journal of Social and Biological Structures, 14, 1. Retrieved from https://www.cmu.edu/joss/content/articles/volume14/Tutzauer.pdf

## Notes

*Summarize*: This paper introduces *affiliation index*, a new measure to characterize affiliation in a two-mode network. As the author nicely argues, this index is two-mode in nature, compared to centrality measures which are one-mode in nature and often borrowed to analyze two-mode networks. The author also introduces a family of affiliation indices and significance tests based on its binomial distribution.

*Reflect*: It is rather shocking that in a 2013 paper, we are still taking about such a simple two-mode network measure. This is not saying the paper is not good, but to say work on two-mode network analysis is surprisingly thin.

I see this family of measures directly applicable in my research settings. Maybe someone already implemented them in some tools. Will find out more.

## Highlights

This article introduces a family of affiliation measures that captures the extent of actors‘ affiliations in the network. (p. 1)

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). (p. 2)

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). (p. 2)

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. (p. 2)

Mathematical Preliminaries (p. 2)

The networks in this article are loopless and undirected. (p. 2)

How Should We Measure Affiliation? (p. 3)

The affiliation index of an actor should measure the extent to which the actor is fully integrated into the structure of the affiliation network—how enmeshed the actor is, by virtue of the actor‘s shared events, relative to other actors. (p. 3)

Affiliation index (p. 3)

The hypergraph also shows that the degree of an actor does not necessarily correspond to how affiliated the actor is with other actors. (p. 5)

There are several ways in which one might assess an actor‘s affiliation using shared events. Their number, size, and multiplicity could be taken into account. (p. 5)

we define actor v i‘s affiliation index, symbolized α(v i), by the equation: )( )(μ )( ii i vd v v . (p. 6)

As a measure of affiliation, α is not perfect. (p. 6)

In fact, one disadvantage of the measure is that it is somewhat ―coarse.‖ (p. 6)

Because the present measure depends primarily on only an actor‘s maximum mutual affiliation without taking into account other mutual affiliations, actors D and F receive the same scores with the present measure whereas a finer-grained measure might distinguish between them. (p. 6)

The above analysis suggests that we might consider the present measure merely the first in a family of measures, what we might call the first-order affiliation index and symbolize α 1. If we let μ i be the ith largest mutual affiliation an actor has, then we would rewrite Equation 1 as α 1= μ 1/ d (if the actor is understood). Moreover, a reasonable second-order affiliation index would be α 2 = (^{1}⁄_{2})[(μ 1 + μ 2)/d]. In general, for k = 1, … , n – 1 we define the kth-order affiliation index for an actor to be the sum of the actor‘s k largest mutual affiliations all divided by kd (dividing by k normalizes the measure to a 0-1 scale). (p. 7)

In practice, it is probably not necessary to compute affiliation indices of all orders. In fact, first-order affiliation may be sufficient in many applications. (p. 7)

Sampling Distribution of First-Order Affiliation (p. 8)

Statistical Model (p. 8)

Under the assumption that the elements of the affiliation matrix are generated independently, many of the variables of interest will have a binomial distribution. (p. 8)

Derivation of the Distribution (p. 9)

An Empirical Example (p. 12)

In this section, we apply the first-order affiliation index and its sampling distribution to an empirical, two-mode network. (p. 12)

the affiliation matrix of 20 corporate directors (the actors) and their memberships in social clubs and various corporate, museum, and university boards of directors or trustees (the events), as reported by Barnes and Burkett (2010). (p. 12)

Examining the diagonal of Table 5, we see that the directors attended anywhere from a low of two events (Chewning and Cushman) to a high of nine events (Livingston and Oates). But attending many events does not mean that these events are shared with another actor. (p. 15)

Using Equations 14 and 15 to compute significance levels, one finds that none of the actors are significantly less affiliated than expected by chance. Five actors, however, are significantly more affiliated than expected by chance (p. 15)

Note importantly that, as in the hypothetical network of the previous section, having a high degree does not necessarily mean that the actor will be highly affiliated in the network. (p. 16)

the first-order affiliation scores fail to match the degree centralities as well as the other three centrality measures. In fact, as Table 7 shows, first-order affiliation is actually negatively correlated with the centrality measures for this network, a possibility remarked upon previously, and one that demonstrates that centrality and affiliation are quite different ideas. Whereas central actors may attend many events, or be between or close to other actors and events, affiliation specifically takes into account the nature of the shared events that actors have. Measuring affiliation is thus a purely two-mode idea. (p. 16)

Discussion (p. 17)

This article has introduced a family of affiliation indices for two-mode networks having a bipartite representation. One of these indices—first-order affiliation, which is based on the largest proportion of events that an actor has in common with another actor in the network—was examined for its sampling characteristics. (p. 17)

Affiliation networks have wide applications in the social sciences, and, because of their special structure relative to traditional, one-mode networks, new tools and measures can fruitfully be developed—tools and measures that take advantage of two-mode information. Of particular interest is the way that actors are affiliated with other actors by virtue of their mutual attendance at events. (p. 17)