The Mysterious Equilibrium of Zombies and Other Things Mathematicians See at the Movies
SAMUEL ARBESMAN
If you were a normal moviegoer sitting through the opening credits of “Casino Royale,” you might notice the snappy visuals, or you might begin dwelling on how this new Bond is a lot less refined than the previous ones, what with all the brawling and fisticuffs.
On the other hand, if you were a mathematician, you would also be noticing that the animated pattern of playing-card clubs, as they hypnotically subdivide on screen, starts to look an awful lot like a fractal.
In any decade there are really only a handful of movies about math (“Proof ” comes to mind, as well as “A Beautiful Mind”), but a surprising number of movies that end up embodying math, even if it’s accidental. “Six Degrees of Separation” is based on the math of social networks. Thrillers have a special propensity for edgy twists on game theory. And what is a disease-outbreak movie if not an illustration of mathematical epidemiology, with puffy suits? To see movies through their math, sometimes, is to watch a whole different drama.
In the opening minutes of the new film “Harry Potter and the Half-Blood Prince,” the Millennium Bridge—an elegant modern footbridge in the center of London—is sensationally destroyed. While casual watchers might recognize this as a nice point of reference to the real world, those who think about math are actually bothered by it, since the bridge buckles while moving up and down.
The Millennium Bridge did indeed have a dangerous wobble to it, but not that one. In 2000, the then-new bridge had to be closed due to a spontaneous side-to-side swing that surprised the engineers who had designed it, and continues to intrigue mathematicians. One explanation, put forth by Steven Strogatz (my graduate school advisor) along with some colleagues, involved the spontaneous synchronization of human behavior, where a slight initial wobble is amplified as people unconsciously begin walking in synchrony to keep their balance, which in turn causes even more people to walk in unison. The most recent theory, proposed by John Macdonald, actually eliminates the need for people to adjust their pace, at least when the bridge initially begins to move.
The bridge has since been modified to prevent the wobbling. And thus far, no math has been used to model the precise forces required for dark wizards to destroy the bridge.
In the final scenes of “The Dark Knight” (spoiler alert!), the Joker gives the following choice to the passengers of two ferries: they can either blow up the other boat and save themselves, or themselves be blown up. If no one decides within a certain amount of time, both ferries are destroyed.
The typical moviegoer pretty much thinks one thing: Batman better show up now. But the mathematician immediately recognizes the Joker’s trap as a variation on the classic problem of the prisoner’s dilemma, where two individuals, each isolated in a prison cell, are given a choice: betray their friend and go free, or cooperate by saying nothing, and be given a short prison sentence. If each betrays the other, however, they will get a longer prison sentence.
This seminal problem in game theory has an important property: while cooperation is a more socially beneficial strategy, it is actually a more “stable” strategy for each person to betray the other, since this makes each better off independent of the whims of his friend. This behavior is known as a Nash equilibrium and is named after John Nash, well-known from the more obviously mathematical film, “A Beautiful Mind.”
Characteristically, the Joker is playing a warped variant of the game, in which cooperation is a very bad choice, since everyone will die. Of course, as science blogger Jake Young has noted, the ferry passengers must take into account something that most participants in the game could safely ignore: the nontrivial probability that they will be saved by a superhero.
You think you know what zombie flicks are about: the undead’s insatiable desire for human flesh. True, but they’re also about epidemiology. One zombie bite infects a healthy person, who infects others, and so on until the few healthy humans left are holed up in Wal-Mart with a shotgun, aiming for the head.
The problem of zombies intrigued Philip Munz of Carleton University and his colleagues at the University of Ottawa, who recently wrote a scientific paper quantifying various properties of zombie epidemics. Standard modeling techniques for disease outbreaks weren’t quite sufficient, the authors found. “The key difference between the models presented here and other models of infectious disease,” they wrote, “is that the dead can come back to life.”
After a thorough, if tongue-in-cheek, analysis, the authors found that the optimal method for halting such epidemics involves killing zombies early and often—the rare scientific paper that satisfies both the splatter-film aficionado and the Centers for Disease Control.
In the film “Six Degrees of Separation,” Stockard Channing’s character at one point makes the following observation: “I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everyone else on this planet. The President of the United States, a gondolier in Venice, just fill in the names.”
Back in the 1960s, the psychologist Stanley Milgram attempted to measure the number of connections separating people in Nebraska and Kansas from those around Boston, and found a typical distance of about six steps. Since then, this concept of a tightly connected world has captured the public imagination, and in math and science has spawned a large field known as social network theory. Jure Leskovec and Eric Horvitz, two computer scientists, recently calculated that, at least in online networks (they used MSN Messenger interactions), the number of steps separating any two individuals on average is about 6.6.
Of course, this area of research is also applicable to Six Degrees of Kevin Bacon, the game in which players compete to link any given actor to Kevin Bacon, through mutual co-stars, in the minimal number of steps. When the calculations are done, it turns out that Bacon is actually not the most connected actor. Dennis Hopper is, with Harvey Keitel running second. Kevin Bacon is 507th.
Speaking of Harvey Keitel, at the end of the movie “Reservoir Dogs” (another spoiler!), Keitel and two other main characters find themselves in a triangle, all pointing guns at one another. This situation, known as a Mexican standoff (and borrowed from “The Good, the Bad, and the Ugly”) is the province of an area of social network analysis known as balance theory. Which social relationships are stable? Well, not this one.
If you and someone else hate the same third person, but like each other, balance theory says you’re golden—all three can persist without changing their opinions. On the other hand, if all three of you despise the others, it’s an unstable triad, as well as a wildly common plot point for crime movies. While there are numerous resolutions—one person changes his preference toward another, a relationship tie is cut—another route back to stability, albeit a messy one, is the gunning down of at least one person. Presumably not a mathematician, who would already be out of there.