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The Small-World Problem:
Six Degrees, Friendster, Mr. Micawber and Kevin Bacon
I. A Problem Defined
Virtual communities sites like Friendster and MySpace are based on the construction of social networks of mutual acquaintances and parley their unusual popularity from the fact that one can, with enough time and effort, create a virtual-visual web of friends surrounding oneself; those friends have virtual-visual webs extending out from them as well, and the webs continue into what is in theory an ever-expanding horizon until the webs encompass the entire world. Basically, it is the friend-of-a-friend-of-a-friend idea developed, due to the nature of the medium, to a seemingly endless number of possibilities for interconnections. Almost anyone would readily agree that contemporary social networks are complicated things, but with this tool, they have the potential to become even more complicated still.
One truly interesting phenomenon that occurs in such virtual environments- and indeed, has served as a way in which Friendster and the other sites market themselves to potential users- is that a person can utilize them to discover a mutual acquaintance with someone else. This is essentially why these network sites are innovative developments; it is what separates them from a run-of-the-mill earlier generation dating service or chat room. In a more traditional, non-electronic environment, the story, which has a ring of urban myth to it, is told of two individuals from different regions of the United States chancing to meet in a foreign country, where they become acquainted and strike up a conversation and discover to their amazement that they share a mutual friend. Perhaps they were even both present at this friend’s wedding years before, but had not met one another at the time. And now they find themselves a long way from home, surrounded by strangers- indeed, essentially strangers to one another- but with this intermediary person, the friend of both, serving as a social link binding the two of them inexorably together. The two individuals perhaps smile and comment that it is a “small world,” and this is where the idea gets its name- the idea that any one person in the world can be reached through a socially constructed network of friends, a chain of acquaintance, is referred to as the “small world problem.”
The idea substantially entered the popular idiom with the publication and subsequent popularity of John Guare’s play, Six Degrees of Separation (1990). The play was a hit on Broadway and was made into a commercially and critically successful film starring Will Smith, Stockard Channing and Donald Sutherland in 1992. In the film version, Stockard Channing’s character Ouisa and her husband (Sutherland) are bilked by a young charlatan (Smith) who pretends to be the son of Sydney Poitier; they are charmed by his conception of the world’s interconnectedness across the racial and socio-economic lines which seem so deeply entrenched in their insular, self-absorbed world of wealth and ease. Ouisa says at one point in the conversation:
I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet.
It is an intriguing idea, of course, but whence did it come? It turns out that this popularized idea, one of the central ideas of this play and movie, which would immediately enter the popular idiom as a sort of urban legend cum parlor trick, is based on scientific work begun in the 1960’s at Harvard by Stanley Milgram and at MIT by Ithiel de Sola Pool.
Milgram sparked scientific interest in the Small World Problem with an experiment and the ensuing publication of an article in the journal Psychology Today in 1967. Milgram starts out his paper by elegantly describing the problem itself:
The simplest way of formulating the small-world problem is: Starting with any two people in the world, what is the likelihood that they will know each other? A somewhat more sophisticated formulation, however, takes account of the fact that while person X and Z may not know each other directly, they may share a mutual acquaintance- that is, a person who knows both of them. One can then think of an acquaintance chain with X knowing Y and Y knowing Z. Moreover, one can imagine circumstances in which X is linked to Z not by a single link, but by a series of links, X-A-B-C-D…Y-Z. That is to say, person X knows person A who in turn knows person B, who knows C… who knows Y, who knows Z.
The basic aim of Milgram’s experiment is to determine “given any two people in the world, person X and person Z, how many intermediate links are needed before X and Z are connected?” The answer that Milgram comes up with is extremely surprising, and quite counterintuitive, but first, one must know about the experiment itself, for like nearly everything Milgram did (as in his classic Obedience to Authority study), it is a work of sheer genius.
Milgram decided to use the mail as the vehicle for his experiment. First, he selected a target person living in Massachusetts, a stockbroker who lived in Sharon and worked in Boston. Milgram then chose two remote places off the top of his head- he selected Wichita, Kansas, and Omaha, Nebraska, which to a scientist in Cambridge, Massachusetts, seemed impossibly remote. He solicited volunteers in these cities and sent them a package that they were to get to the target individual in Massachusetts- the banker- via mail in as few steps as possible. The rules were simply that they had to mail the package to one other person and that they had to be on a first-name basis with the person to whom they were mailing the package. If they knew the man in Massachusetts, then they could just mail it to him directly (turned out that none of the volunteers did!), otherwise, they had to think of someone they knew personally who might either know the target individual or might know someone else who could get the package closer to its destination (and so serving as intermediaries in the chain); all participants who received and sent the package would record their names in a log that was traveling with the package itself. After 15 attempts to get the package to the target, that is, 14 intermediary chains, the chain was considered dead and no further attempts would be made. Again, the experiment is a work of sheer brilliance, but there was serious doubt that it would create results. Could it possibly work? Sociologists whom Milgram polled predicted that it would take hundreds of intermediaries before there was any hope that the package could reach its target, if at all.
But the results turned out to be truly surprising to everyone involved. In one case, the package reached its destination in a chain of only two intermediaries. When all the results were in, the median number of passes between individuals in successful chains in which the package actually reached its destination was five. Thus, the phrase “six degrees of separation” was born. Also surprising was the fact that in many cases in which the package chain died out before it reached its destination, it wound up within a rather small radius of the target; in some cases, it died out within the very neighborhood of Sharon in which the target individual lived. Also of interest was the fact that certain individuals were habitual “chain closers”- that is, time and again, even though the packages had started with different people in Omaha and Wichita, they passed through their hands on the way to their destinations. These habitual “chain closer” were social nodes with large numbers of connections; these very same social nodes- seemingly centers of the social electronic universe, exist on Friendster and its online clones.
Milgram’s work was entirely practical in basis, but the mathematician Ithiel de Sola Pool, who was working at MIT at the time of Milgram’s experiment, formulated an important mathematical model that dealt with the small-world problem. His work is rather complicated, but here is a summation. Basically, he needed to determine how many individuals the average person knows. He asked a cross-section of individuals, men and women, to record all the names of individuals whom they met in the course of a 100-day period. The average number this pool of individuals had recorded after the 100 days was around 500. Using some simple mathematics, Pool was able to determine using national population figures that the chances of any two people knowing one another is one in 200,000. But most surprising, the chances that any two people in the United States could be linked up through two acquaintances- that is, three degrees of separation- was more than 50%. This means that for any person in the entire country, it is more likely than not that he or she knows someone who in turn knows me.
So why does anyone care about this stuff? As Milgram and Watts point out, when social networks break down, the effects can be far-reaching and profound. Milgram argues that the Dark Ages in Western Europe were brought on by the breakdown of communication between Medieval cities. This breakdown caused a decay of the very social fabric, which in turn destroyed the social networks that had been set up under the Romans for trade, communication and agriculture. As time passed, the snowball effect intensified, and individuals became isolated and constricted within smaller and smaller groups. The monasteries themselves- where knowledge was by-and-large preserved and handed down to future generations, eventually reaching us- were places of retreat from the stir of social life- and thus, even further removed from these already weakened social networks. Watts devotes entire chapters of his recent book Six Degrees: The Science of a Connected Age to the real-world consequences of problems with social networks; he cites examples such as the Asian stock market crises of the late nineties and several recent examples of outbreaks of epidemic illness. So social networks, on display in microcosm within the realm of Friendster and the countless other services like it, are important things indeed.
One might express surprise upon learning that the theory underpinning both the parlor game and the slew of virtual communities sites on the Internet is based on serious scientific research going back to the late 1960’s. But the idea of constructions of society in which an individual exists within a bewildering array of social networks is, in our opinion, even older and more storied than this. These types of social networks- intricate, complex, unknowable, vast- operate within the realm of nineteenth-century fiction. Specifically, they are formulated and developed with the rise of the nineteenth-century urban novel, as typified by Dickens, Trollope and Thackeray, and receiving a twentieth-century treatment by Joyce in Ulysses. The most complex and far-reaching (and generally most widely critically-acclaimed) urban novel of the nineteenth century is Dickens’ Bleak House (1853). Here is presented a social network complex in the extreme, with a seemingly never-ending stream of richly-drawn characters existing within the enormously complicated fabric of nineteenth century London. Of particular interest is the nature of the characters’ interactions. They operate within a world of removes; that is, they interact through intermediaries, establish extended acquaintanceship chains, experience within those chains chance meetings and encounters and find in several cases that their circle of friendships is nothing more than an extended loop which circles back on them (sometimes with dire dramatic consequences, since, after all, this is Dickens). The social network itself becomes more the focus of the plot than any one character within it, though each character acting in his or her own peculiarly Dickensian extreme of behavior affects the ebb and flow of the entire structure of the plot itself.
So it is with the virtual communities, which encourage users to sign up and provide their personal information (including, in some cases, an email address or some other contact outside of the network) in hopes that they will run into a long-lost friend or perhaps discover that they themselves are the missing link- Milgram’s Y- connecting two of their friends- Milgram’s X and Z. And just looking at these sites from the limited view of anecdotal evidence of our own friends, it seems that they are at least somewhat successful in the attainment of this goal. And just like in Dickens’ Bleak House and in Milgram’s experiment, something about the modern world- a condition of modernity, if one wants to use the language of literary theory- is allowing or forcing these interactions to occur. In Dickens’ case, it was the city itself- the rise of “urban-ness,” close living, cramped conditions, which brought about the possibility of chance encounters and the extension of social networks on a scale never before imagined. In Milgram’s experiment, the mail served as the device of connection, the condition of modernity allowing for the further extension of acquaintanceship chains. With Friendster and its clones, the Internet, itself one of the most complex networks ever constructed, is serving as the conduit. Social networks in agrarian, pre-technological societies just do not have the potential to develop the same complexity and scope. They are more like the social networks of the Dark Ages, composed of isolated pockets of individuals and monastic groups that do not link up, and so, are by their natures not as vital.
Can everyone in the world be connected up within six degrees, as Ouisa claims in the Guare’s play? Are you only a few degrees removed from Dick Cheney? Watts raises the issue of social networks versus social groups in his book. He wonders if models like Pool’s are capable of holding up in real life since all societies are by nature fractured into social groups determined by factors like race and socio-economic background. As Milgram puts it in his 1967 paper:
So the big obstacle one runs up against is the problem of social structure. Though poor people always have acquaintances, it would probably turn out that they tend to be among other poor people, and that the rich speak mostly to the rich. It is exceedingly difficult to assess the impact of social structures on a model [like Pool’s]. If you could think of the American population as simply 200 million points, each with 500 random connections, the model would work. But the contours of social structure make this a perilous assumption, for society is not built on random connections among persons but tends to be fragmented into social classes and cliques.
But herein lies the most important feature that we must consider when we look at sites like Friendster and assess their potential. Is it stretching logic a bit to assume that they herald a new era of social interconnectedness which overcomes the societal barriers of which Milgram speaks? As more and more people gain access to computers- as their price decreases (unlike almost any other consumer product)- and it increasingly becomes the case that to function normally in society, one must interact with the Internet in daily life, will social networks expand, grow larger, and even perhaps, approach something akin to the random distributions of which Milgram speaks? Again, Friendster would only be an embryonic development along the road to such interconnectedness, but it might nonetheless be argued at some point in the future, an important one.
The parlor game with which everyone is now familiar and which popularized the small-world problem as nothing else before it was devised in 1996 by two graduate students in computer science at the University of Virgina, Brett Tjaden and Glenn Wasson. Utilizing the ideas of Milgram, Pool and other researchers who had contributed to small world and social network research, they theorized that any actor or actress in the history of film could be linked up with Kevin Bacon in a small world chain of no more than six degrees. To illustrate their point, the pair created the Oracle of Kevin Bacon, which utilizes the massive Internet Movie Database to assign every actor or actress living or dead, obscure or prolific, with a Bacon number. And it works! Almost every actor, living or dead, has a Bacon number of no more than three, with the vast majority of those being assigned a two. Try as I might, I simply could not find an actor with a Bacon number higher than two. I tried to think of random character actors which I assumed could stump the system- Victor Mature, Ida Lupino, Celeste Holm, Judy Holliday, Ray Bolger, and I even tried the fabled Greta Garbo, and all of them, even Garbo, turn out to have Bacon numbers of two. And just like in Milgram’s experiment, there are nodes that serve as conductors to Bacon. In particular, the actor Eli Wallach, a prolific character actor with a career stretching back to the fifties and who managed to work, in his long career, with seemingly everyone, appears with Bacon in Mystic River (2003). Also, the film JFK, which, like other Oliver Stone films, had an enormous cast of extras and famous actors in cameo appearances, and featured a performance by Kevin Bacon, serves itself as a concentrated node through which many Bacon connections are made, particularly through the actor Jack Lemmon, who also appeared in the film. So it turns out that the star of such perhaps forgettable films as Footloose, The River Wild, The Air Up There and Tremors is only one remove away from the legendary Bette Davis, which only goes to show that we do, in fact, live in a small world after all.
Milgram, Stanley. “The Small World Problem.” Psychology Today. 2, pp 60-67, 1967.
Steele, Bill. “Kevin Bacon Shows the Way to a Smaller World.” Cornell Chronicle. June, 1998.
Tjaden, Brett and Wasson, Glenn. “The Oracle of Kevin Bacon at Virginia.” <http://www.cs.virginia.edu/oracle/>
Travers, Jeffrey and Milgram, Stanley. “An Experimental Study of the Small World Problem.” Sociometry, 32(4), pp 425-443, 1969.
Watts, Duncan J. Six Degrees: The Science of a Connected Age. Norton, 2003.
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