KEVIN BACON AND GRAPH THEORY

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Mar 2004 by Hopkins, Brian

The Oracle is fast and user-friendly. It has a generous system for dealing with misspellings, offering likely candidates close to names not in the database. It will find the distance between any two actors, not just to Kevin Bacon. The Oracle provides links to the IMDb pages for actors and films and does several summary computations, as discussed below.

4 CLASSROOM PRESENTATION: WHO'S THE CENTER?

Most students have heard of six degrees of separation, and many are familiar with the Kevin Bacon Game or at least have seen a recent commercial based on the idea [8]. After I have explained Milgram's work and the set-up for the cinema graph, we spend several minutes on the Oracle looking up the distances of various actors from Kevin Bacon. Students are surprised that the distances are so small and usually cannot come up with an actor more than 3 away.

But why, a student usually asks, of all possible actors, is it Kevin Bacon? The game's creators, Pass, Turtle, and Ginelli, explain that they had watched Footloose one snowy day. That night, there was a television advertisement for the upcoming film The Air Up There, also starring Kevin Bacon.

Somehow it occurred to us that Kevin Bacon had been in so many different types of movies, you could connect a lot of unlikely people together through his work. Well, that idea took off. We found we could actually get anybody back to Kevin Bacon in a few steps. There could only be one explanation-Kevin Bacon was at the center of the entertainment universell, pp. 13-14]

Is Bacon really the best choice for the game? What would it even mean to be the best choice?

We want a way to measure an actor's centrality in the cinema graph. Students often suggest using the number of co-stars, i.e., how many actors are at distance 1. For example, there are 1703 actors distance 1 from Kevin Bacon (so the full cinema graph has 1702 other edges connected to the Kevin Bacon vertex in Figure 1). There are 1527 actors distance 1 from Wesley Snipes. (All data were supplied by the Oracle in December 2003; the IMDb is updated frequently, so numbers used here may have changed by the time of publication.) So you might conclude that Kevin Bacon is better connected in the cinema graph than Wesley Snipes. But Snipes' fewer co-stars connect to more actors than Bacon's, that is, there are more actors distance 2 from Snipes than from Bacon. Likewise, there are more actors distance 3 from Snipes than from Bacon. How can these data be incorporated?

The weighted average does this well; the computation for Kevin Bacon is shown in Figure 2. (Notice that there are actors up to distance 8 from Kevin Bacon, so "Six Degrees of Kevin Bacon" is a misnomer.)

So the average distance to Kevin Bacon is 2.948. The complete data for Kevin Bacon, Wesley Snipes, Anthony Quinn, and Rod Steiger are given in Figure 3, along with the weighted averages. Notice that Snipes' weighted average is 2.923, slightly lower than Bacon's. Thus Wesley Snipes is, on average, more closely connected to other actors, so he is a better center.