KEVIN BACON AND GRAPH THEORY

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

As with Bacon and Snipes, there's a similar relationship between Anthony Quinn, who has more co-stars, and Rod Steiger, who has a lower weighted average. It may surprise you that Rod Steiger is the best connected actor by this measure. he was both a leading man and character actor in 50 years of films, performing in musicals, dramas, comedies, horror films, independent U.S. films, and European films. There are better known actors, but they have not performed as long in so many genres. The top 1000 best connected actors are listed on, the Oracle; Kevin Bacon just misses that distinction.

As you compute the weighted averages, you see that the denominator in all four cases is 606,429. The four actors are all connected to each other, so they are all connected to the same large set of actors. But the IMDb includes some 620,000 actors; where are the rest? Not all actors connect to the Hollywood mainstream. Type "Olga Holts" into the Oracle, for example, and it will report a distance of infinity from Kevin Bacon. The cinema graph has a large connected component of 606,429 vertices with the remainder falling into several small connected components. Most components consist of only a few vertices, but Olga Holts' vertex is part of a component of 29 corresponding to Estonian silent film actors of the 1920s.

5 OTHER ACTIVITIES AND STUDENT RESPONSES

In order to reinforce student understanding of the cinema graph's structure, I ask them in homework or a collaborative project to draw the connected component that includes Olga Holts' vertex. The Oracle will list all of an actor's co-stars, and switching over to the IMDb provides complete data on the four films connecting these 29 actors. The drawing consists of four overlapping complete graphs.

The IMDb is updated regularly, so the cinema graph is dynamic. Homework or project questions addressing this are "Can a deceased actor's Kevin Bacon number decrease? If so, how low can it go?" In more advanced settings, I explain that the diameter (the maximum distance) of the cinema graph is 14 and ask "Over time, do you think the diameter of the cinema graph decreases, stays the same, or increases? What factors are relevant to the diameter?" One student suggested that the diameter decreases, explaining that there are new films (which can decrease the diameter) without new actors, while new actors (which can increase the diameter) can only be introduced in new films. In fact, the diameter of the cinema graph has decreased over time [7].

Also, I offer extra credit for finding an actor with (finite) distance 5 or more from Kevin Bacon (with no credit for students who turn in names from the same film). The IMDb allows for searches by genre, language, and decade, so this is more about database work than obscure film knowledge. Most students can find an actor with distance 5 from Kevin Bacon; recently a student found one of the two Kevin Bacon 8's!

Student response has been very favorable. Graph theory students enjoyed exploring a "real graph" of significant size. Working with a graph larger than one I could draw on the board helped them realize the subtlety of various definitions and the need for efficient search algorithms. Finite mathematics students liked the application of weighted averages to deciding which actors are better connected. The discussion of the Kevin Bacon Game, and basic graph theory in general, engaged several finite mathematics students who had not previously participated at a high level. One such student said, "This isn't math!" I was very happy for her to see mathematics other than formulas and variables.