Friendster, the hugely popular "social networking"/hookup engine, has attracted attention from venture capitalists, lonely twenty-somethings, and at least one friend of mine looking to add a little extra oomph to a breakup by defriendsterizing her ex. It’s also attracted fakesters, people who have decided to enliven the site by adopting identities like Giant Squid, Bat Boy, or Patty Hearst. People like the fakesters; Giant Squid had 335 friends tied to his account. The Friendster corporate team doesn’t care for them nearly as much, and they’ve been angering some users by shutting down fake accounts left and right. Fakesters like "Ecstacy" and "Tootsie Roll" have entered the graveyardster; innocent non-fakes have been caught in the purge. One particularly vulnerable group, apparently, has been DJs; they’re sociable, well-liked, and oddly named, three symptoms of fakester behavior. It’s not easy being popular. Researchers discovered social networks a while ago. Sociologist Stanley Milgram did a series of experiments in which subjects were asked to deliver letters, via a chain of acquaintances, to complete strangers across the country. Milgram’s research may have been flawed (although Columbia University’s recent Small World experiment has provided some confirmation), but it provided one catchy idea: six degrees of seperation. Milgram put his results through a strainer and decided that strangers (from similar racial and social backgrounds, admittedly) who lived across the country from one another across the country were linked by an average chain of just six acquiantances. And some strangers were more popular than others; in an article about a truly remarkable specimen of stranger-binding, Malcolm Gladwell notes that

Some [subjects] went through college acquaintances. Some sent their packets to relatives. Some sent them to old workmates. Yet in the end, when all those idiosyncratic chains were completed, half of the packets passed through the hands of Jacobs, Jones, and Brown. Six degrees of separation doesn’t simply mean that everyone is linked to everyone else in just six steps. It means that a very small number of people are linked to everyone else in a few steps, and the rest of us are linked to the world through those few.

The six degrees of separation theory posits that just a few highly connected nodes tie together networks. It seems to be true, too; in seemingly disparate networks like the neural map of nematodes, American power grids, or P2P filesharing systems, a few connectors make all the difference in tying things together. Malcolm Gladwell knew Lois Weisberg (and Lois Weisberg knew everyone from Harold Washington to Arthur C. Clarke); Friendster’s users knew the Tootsie Roll (and Friendster’s rules imply that Tootsie Roll therefore "knew" everyone else). You can examine a map of the Western U.S. power grid and figure out which power station bear the most connective responsibility. But how does a researcher figure out who the connections are in a more abstract, social network?

In some cases, they just make it up. The Kevin Bacon game rests on the fact that practically every actor who ever appeared in a feature film made in any country, in any language, since the beginning of motion pictures, can be linked to Kevin Bacon in six steps or less. (The keepers of the Oracle of Bacon used to record people who found actors that required seven steps in a Hall of Fame.) A good map of film roles and some skill with databases are all that’s needed to demonstrate that, say, Elaine Garreau has a Kevin Bacon number of 3 (Garreau to Frank Langella to Kevin Bacon). But over a thousand actors are better centers of the film universe than Bacon, who just caught the attention of the Game’s inventors as a young actor who worked frequently and with a great many different actors.

The idea of the Kevin Bacon number was inspired by the idea of the Erdös number, and the selection of Paul Erdös was far less arbitrary. Erdös, a Hungarian itenerant academic, speed freak, and legend in the mathematical community. He was the subject of a two popular biographies and a documentary film. Erdös was modern mathematics’ second most prolific author of papers, and an astonishing proportion of them were significant. Just as importantly, Erdös was a maniac for collaboration, with a staggering 507 coauthors throughout his career (and the number continues to rise as papers are published posthumously).

Like Swift’s fleas, Erdös’ collaborators have collaborators, and so on ad infinitum; six thousand people, including Albert Einstein, are collaborators of Erdö’s collaborators, and thus have an Erdös number of two. Practically the whole of the twentieth century’s most celebrated mathematicians and physicists have small Erdös numbers. Being what they are, mathematicians (particularly graph theorists, who are concerned with networks and work in one of the fields that Erdös published in most frequently) have plotted out a number of interesting facts about Erdös’ social network, making it perhaps the most studied example of a time-stamped (research collaboration) multigraph.

Thanks to that documentary, Paul Erdös had a Bacon number of four. His Erdös number was, of course, zero; barring a career change late in life, Kevin Bacon is unlikely to receive an Erdös number of his own. A handful of mathematicians have finite Bacon numbers, largely through cameo appearances in A Beautiful Mind. To the best of my knowledge, the only working actor with a finite Erdös number is Danica McKellar, whose senior thesis, "Percolation And Gibbs States Multiplicity For Ferromagnetic Ashkin-Teller Models On Z2" (by Chayes, McKellar, and Winn), was published in Journal of Physics A. McKellar’s most noted role was that of girl-next-door Winnie Cooper on The Wonder Years, but she’s been popping up on The West Wing lately and, as Daniel Drezner notes, has played the lead role of a mathematician’s daughter in the West Coast staging of Proof. McKellar’s official website (now defunct) used to offer a column called "Dear Danica", which offered "math tutoring in addition to advice on love, life, and relationships", making her perhaps the coolest human being in Hollywood.

But what of people who aren’t wired into society? William Labov, a scholar of African-American Vernacular English, examined the lives and speech of adolescents in Harlem. He asked children to list their friends, then compared lists. Those who listed people as their friends but were not themselves listed as friends, he discovered, had distinct language patterns from those more tightly bound into the social networks. The chapter of his book that discusses the phenomenon sums it up neatly: "The Linguistic Consequences of Being a Lame."