## Fun Fact - Bacon Numbers

As you probably know, the Bacon number of an actor or actress is his or her distance from Kevin Bacon, where distance is the Erdős number. For example, Kevin Bacon's Bacon number is 0. If an actor works in a movie with Kevin Bacon, the actor's Bacon number is 1. If Wilma works with Betty who worked with Kevin Bacon in a movie, Wilma's Bacon number is 2, and so on. The average Bacon number is 3, and the largest Bacon number thus far is 8. There's a lot of mixing and interplay in movies, and it only takes one movie with a lot of actors to decrease a lot of Bacon numbers across the board.

Whenever I encounter a new concept, I like to step back and examine it across time, a thousand years, a million years. It's really a form of mental illness I suppose, because it shines a spotlight on my insignificance and gives me some serious angst, but that's what I do, so here goes. Your first exercise is to prove that Bacon numbers increase with time. It's intuitive, but pause and see if you can make it formal.

A person's acting career is less than a century, often much less. If Kevin Bacon is acting in 2000, he's not acting in 2100. No one acting in 2100 has a Bacon number of 1, so their numbers are at least 2. In other words, all the actors with numbers of 1 act prior to 2100. If fred has a number of 2, he acts at the same time as an actor with a number of 1. All the actors with Bacon number ≤ 2 act prior to 2200. By induction, all the actors with Bacon number ≤ n act prior to 2000 + 100 × n. Therefore, Bacon numbers steadily increase with time. A thousand years from now, all the actors alive at that time will have Bacon numbers that cluster around 20, assuming a significant fraction (say 20%) of actors have 60 year careers more or less.

Ok, that was a warm up. Do Bacon numbers spread apart with time? A million years from now, will contemporary actors have Bacon numbers that differ by 100, or 1000? That's a harder question. I'm not sure, but I think the answer is no. Perhaps Fred's number is 854 and Barney's number is 981, but all they have to do is appear in a movie together and Barney's number snaps back to 855. At the same time, all the actors close to Barny snap back to 856, 857, and so on. Separation is unstable, like a house of cards, and one movie can tighten things up. The only way to maintain a separation long term is if actors become separated into two groups by some perverse social norm, some form of racism that rears its ugly head in the future. Sadly, we know that this can happen at any time, as though it were hard wired in the brain. Perhaps, as an example, chinese Actors and English actors never appear in movies together, and maybe that persists for several decades, like our segregated baseball leagues prior to Jackie Robinson, but eventually some maverick will cast a Chinese actor and an English actor together, and all the Bacon numbers snap back together. With this in mind, I surmise that the spread of Bacon numbers will not exceed 20, even after a million years. It is always bounded by the diameter of the connectivity graph of the living actors at the time, which is currently 15. If the smallest Bacon number is 900, then take 15 steps, and the largest Bacon number is at most 915.

Here's something that is almost certain to happen however. At some point humanity will fall back into the dark ages, or even another stone age if we can't figure out how to eliminate nuclear weapons. If we stop making movies even for a century, the Bacon numbers stop. After the Renaissance, we can start them up again, but we need to select a new actor as the center of our movie universe. Then some archeologist will publish a paper in a peer reviewed journal:

“After careful translation of the ancient texts, we have discovered that the people of the previous technological epoch played a similar game with actors and connectivity. The center of their network was Kevin Bacon, an actor that we know nothing about. Dr. Stevens, chair of the department of ancient studies at the University of Gorsit, suggests that Kevin Bacon must be a unique and gifted actor, thus earning his central status in the network. However, Dr. Froth, from the math department at Yohon State University, suggests that he was simply in a lot of movies, thus the resulting "Bacon numbers", as they called them, were small and tightly clustered. To help answer these unresolved questions, our team is diligently combing through the archives of that era, searching for a movie with Kevin Bacon.”

This does not address the possibility of humans living forever, or for a very long time. I don't think forever is possible, at least not biologically, and even if you never age, and all diseases can be cured, there are still accidents, disasters, murders, and suicides. But what if humans extend life to thousands of years? Well to begin with, movies are the least of our problems. We must find a way to regulate births, else starvation and disease are inevitable. Perhaps you receive a permit from the government to have a child when somebody dies. Let's say we've solved that problem. Bacon numbers still increase, as per the previous proof; they just increase more slowly. As for the spread, I think it is even less. No matter how long we live, there are no more than 10 billion people on earth, because the planet won't support more than that. So we have the same number of actors, making the same number of movies, but their acting careers are much longer. There are more opportunities to appear in the same movie together, thus the Bacon numbers tighten up. Perhaps the diameter of the graph is no more than 6, which, in turn, is a bound on the spread of the Bacon numbers.