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"Since you cannot do good to all, you are to pay special attention to those who, by the accidents of time, or place, or circumstances, are brought into closer connection with you."

~Augustine of Hippo~

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At family reunions, my mother used to joke about the fact that actress Shirley Jones (of "The Partridge Family" fame) never showed up. Apparently someone had figured out that Jones was a distant relation by marriage, and more than once I heard of plans by one of my cousins to invite her to a reunion just for fun. Whether an invitation was ever actually sent I don’t know, nor can I recall the exact chain of relatives that supposedly connected me to Jones. But I always enjoyed the thought of being related, however distantly, to a celebrity. I imagined showing up at a dinner or award ceremony where I’d be introduced to the rich and famous: "This is Joe Kissell, my third cousin-in-law, once removed." I didn’t suspect it at the time, but if a prominent sociological theory is correct, I may have indirect social connections to millions or even billions of people.

The theory in question, of course, is that of "Six Degrees of Separation"—roughly, the notion that anyone can form a chain of personal contacts leading to any other person, with no more than six links in the chain. Nearly everyone has heard of this idea, thanks to John Guare’s 1990 play "Six Degrees of Separation" and the "Six Degrees of Kevin Bacon" game that became popular in the late 1990s. But what many people don’t realize is that this game has its roots in serious sociological research, and that work is currently underway to establish the validity of the theory scientifically.

Network (of) Associates

The late Stanley Milgram, a Harvard social psychologist, performed an experiment in 1967 to determine the extent of social networks in the United States. In his study, it took an average of six steps for a letter sent to a random resident of Omaha, Nebraska to reach a target person in Boston—using only personal contacts to form a chain. This was the origin of the "six degrees of separation" idea, but Milgram’s small experiment was hardly conclusive, restricted as it was to a very small sample size (and with all participants within the U.S.).

Nearly four decades later, as part of the Small World Research Project at Columbia University, sociologists attempted a modern version of the experiment using email. The project worked like this. You registered on Columbia’s Web site, providing demographic information about yourself and answering a range of questions to help the researchers make sense of their results. You were then presented with the name and location of a target person somewhere in the world. The idea was to send a specially formatted email message to someone you know who you think is closer to the target (in one way or another) than you were. When this next person in the chain received your message, the process repeated until, the researchers hoped, a chain formed all the way from you to the target person. Among other things, the experiment hoped to determine what the average length of such a chain is, providing some statistical validation or refutation of the "six degrees" notion.

Connect Four

The experiment ended after a couple of years, producing some very interesting results (only some of which have been published so far). In all, more than 60,000 people from 166 countries participated in the experiment’s first round. Although over 24,000 chains were started, only 384 of them successfully reached their targets. Most of the unsuccessful chains were broken simply because someone along the way chose not to participate. The high number of incomplete chains skews the otherwise impressive results: the successful chains required an average of only four links. However, when they factored in how long the broken chains most likely would have been, had they been completed, the researchers estimated that people are separated by five to seven degrees—with shorter chains expected between people living in the same country. In other words, there’s apparently nothing magical about the number six, but it does happen to be pretty close to the average number of social links most people have to most other people, at least among the 600 million or so of us with internet access.

One of the main goals of this project was to determine which factors people consider in trying to form associative links. Every time participants selected a new message recipient, a form asked why they chose that person—for example, due to geographic proximity or a shared profession or hobby—as well as what their relationship to the person is (friend, coworker, relative, etc.). The experiment found that participants considered geographic proximity and similarity of occupation to be primary factors in choosing how to continue a chain. And although people most frequently chose friends as the next link, professional relationships were much more likely to lead to a successful chain. I participated in a couple of chains in the experiment’s first round, and although I consider myself relatively well-connected, neither of my chains successfully found its target. But the question is not whether there is a link between any two people; it’s how you go about finding it. Email may or may not be the best discovery method, though it certainly does have a kind of efficiency that casual conversation or letter writing do not.

Supposing it turns out that everyone on the planet is indeed connected to everyone else by no more than six steps, so what? Sooner or later, given the sheer number of people on Earth and the ways in which people form networks, the six-degrees-of-separation theory must become nearly certain mathematically. But does that actually mean anything? Will it lead to world peace, or will it just facilitate the spread of viruses—biological or digital? I don’t know, but I’m sure it will lead to something. Let’s talk about it over lunch. I’ll have my wife’s friend’s assistant set it up with your brother’s neighbor.

How does 'six degrees of separation' work? (Cyber)

Explanation is personal networking, Cornell computer scientist says

Even in the vast confusion of the World Wide Web, on the average, one page is only about 16 to 20 clicks away from any other. But how, without being able to see the whole map, can we get a message to a person who is only "six degrees of separation" away?

A Cornell University computer scientist has concluded that the answer lies in personal networking: We use "structural cues" in our local network of friends. "It's a collective phenomenon. Collectively the network knows how to find people even if no one person does," says Jon Kleinberg, assistant professor of computer science, who published his explanation in the latest issue (Aug. 24) of the journal Nature.

His research is based on a computer model showing that an "ideal" network structure is one in which connections spread out in an "inverse square" pattern. In human terms that means that an "ideal" person in the model would have just about as many friends in the rest of the county as in the neighborhood, just as many in the rest of the state as in the county, just as many in the whole nation as in the state, and so on, as you might find in a highly mobile society.

Kleinberg's answers might have a very practical use in helping to reduce the number of clicks needed when surfing the web, as well as helping to speed up other kinds of networks.

Although Kleinberg has been instrumental in the development of improved search engines for the web, he doesn't see this work as applying to traditional search engines. They already have the "big picture" of the network, he explains, since they work from indexes of the web. Rather, he sees it being useful in the construction of "agents," computer programs that will jump around the web looking for specific information.

It could also apply to the distribution of data over the Internet, where computers called routers must send packets of information on their way toward their destinations without knowing what the state of the network is outside of their own immediate neighborhood.

Kleinberg has shown that a computer algorithm (the basic design for a program) can choose the best way to send a message to a faraway place in a network even if it has knowledge only about the characteristics of its immediate neighborhood. "The correlation between local structure and long-range connections provides fundamental cues for finding paths through the network," he writes in the Nature paper.

Kleinberg's work is a refinement of an earlier study by two other Cornellians, Steven H. Strogatz, professor of theoretical and applied mechanics, and his graduate student, Duncan Watts, now an assistant professor in Columbia University's sociology department.

Strogatz and Watts offered a mathematical explanation for the results of a landmark experiment performed in the 1960s at Harvard by social psychologist Stanley Milgram. The researcher gave letters to randomly chosen residents of Omaha, Neb., and asked them to deliver the letters to people in Massachusetts by passing them from one person to another. The average number of steps turned out to be about six, giving rise to the popular notion of "six degrees of separation," and eventually the "six degrees of Kevin Bacon" game in which actors are connected by their movie appearances with other actors.

Strogatz and Watts created a mathematical model of a network in which each point, or node, is closely connected to many other nodes nearby. When they added just a few random connections between a few widely separated nodes, messages could travel from one node to any other much faster thn the size of the network would suggest. The six degrees of separation idea works, they said, because in every small group of friends there are a few people who have wider connections, either geographically or across social divisions. They also showed that such cross-connected networks exist not only between human beings but also in such diverse places as computer networks, power grids and the human brain.

But Kleinberg has found mathematically that the model proposed by Strogatz and Watts doesn't explain how messages can travel so quickly through real human networks. "The Strogatz-Watts model had random connections between nodes. Completely random connections bring everyone closer together," Kleinberg explains, "but a computer algorithm would have only local information. The long-range connections are so random that it [the algorithm] gets lost."

So Kleinberg designed a model in which nodes are arranged in a square grid and each node is connected randomly to others but with "a bias based on geography." As a result each node is connected to many nearby, a few at a longer distance and even fewer at a great distance -- the "inverse square" structure. "This bias builds in the structural cues in my long-range

connections, and it's the bias that is implicitly guiding you to the target," Kleinberg explains. "In the Strogatz-Watts model, there is no bias at all and, hence, no cues -- the structure of the long-range connections gives you no information at all about the underlying network structure."

The sender of a message in this system doesn't know where all the connections are but does know the general geographic direction of the destination, and if messages are sent in that direction, Kleinberg says, they arrive much faster than they would by completely random travel.

Kleinberg explains, "The Watts and Strogatz model is sort of like a large group of people who know each other purely through electronic chat on the Internet. If you are given the user ID of someone you don't know, it's really hard to guess which of your friends is liable to help you forward a message to them.

"The inverse square model is more like the geographic view of Milgram's experiment -- if you live on the West Coast and need to forward a message to someone in Ithaca, you can guess that a resident of New York state is a good first step in the chain. They are more likely to know someone in the Finger Lakes region, who in turn is more likely to know someone in Ithaca and so forth. Knowing that our distribution of friends is correlated with the geography lets you form guesses about how to forward the message quickly."

The geographic information on the grid, he adds, is an analogue of the social connections between people. Just as nodes on his simulated network choose the correct geographical direction to send a message, so humans may choose a social direction: A mathematician who wants to send a message to a politician might start by handing it to a lawyer.

On the other hand, he says, "When long-range connections are generated uniformly at random, our model describes a world in which short chains exist but individuals, faced with a disorienting array of social contacts, are unable to find them."


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