MATH MODEL EXPLAINS “SIX DEGREES” PHENOMENON

August 25, 2000

SCIENCE & ENGINEERING NEWS

Ithaca, N.Y. — We all know it’s a small world: Any one of us is only about six acquaintances away from anyone else. 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 than 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.” The paper in Nature is titled “Navigation in a Small World.”

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