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Six Degrees of Separation
Through just five or six intermediaries, you could be linked to millions of others. It is the notion behind what has been dubbed the small world effect.
It has happed to most of us, and it's virtually guaranteed to happen at again - especially if you attend a social gathering. Whenever you have people eager to talk with one another, the chances are that some will find that they have friends and acquaintances in common. It is, as they say, a small world.
It may be a common enough experience, but the so-called small world effect is turning out to have some pretty big consequences. In the last 18 months it has become one of the hottest subjects in science. Now some believe it could revolutionize the way we think about everything from economic crashes to globalization.
The story of how an apparently trivial social phenomenon turned out to have far from trivial implications has its origins in a bizarre experiment carried out over 30 years ago by psychologist Stanley Milgram.
Milgram was trying to uncover the connections that lurk in our networks of friends and acquaintances, and hit upon a novel way of revealing them. He recruited people in various US states and sent each of them a package, together with some instructions.
These revealed that the packages were actually intended to two people picked by Milgram, who gave their names and some vague clues about where they lived, their occupation and age. What he did not give, however, was a precise postal address. The participants were then told to send the packages to whichever of their acquaintances they judged most likely to know the targets personally and be able to make the final delivery.
Keeping track of the postings, Milgram made the stunning discovery that the packages typically reached the two target people after passing through the hands of just five other people. Later experiments produced similar results, making the conclusion inevitable. It seems that, on average, everyone in America from arms dealer to zoo keeper can be connected to everyone else via a chain of just five or six intermediaries.
It is a result that becomes more bizarre the more you think about it. Sociologists estimate that we each typically have around 300 or so acquaintances - people we're on first-name terms with. That suggests we're just one hand-shake (or email) away from 300 people, two away from 90,000, three away from 27 million and so on.
Viewed this way, the real surprise about Milgram's research is that it takes as many as five or six handshakes to connect every American to every other. An average of just four should suffice to connect up to 250 millions inhabitants of the US.
But there is a big assumption in this quick calculation: that our 300 friends are randomly spread throughout the population so that every American is likely to know, say, Alan Greenspan as Al at the corner store. But the fact is that our friends tend to fall into cliques: people who have similar levels of education, interests and opinions.
This, however, just makes Milgram's findings even more baffling: for if all our friends were confined to such rigid cliques, we would hardly ever discover we have friends in common. Each American, for example, would then be separated by an average of almost one million handshakes?250 million divided by 300 from each other. By that reckoning, Milgram should have died long before any of his packages reached their targets.
There is clearly something odd going on here. Our networks of friends are not randomly spread across society. Yet they still allow us to be linked to each other via few intermediaries, so that we often end up discovering "It's a small world". How do they do it?
It was this that intrigued Duncan Watts, in 1996 still a graduate at Cornell University. Watts had been working on a nice, solid doctorate about the chirps of lovelorn crickets. But he had run into a problem: how do the crickets fall into step so quickly? Was each listening to all his fellow crickets, or just to his closest neighbors?
Then Watts remembered a funny bit of folklore that his father had told him: that every American is just a few handshakes away from knowing the president of the United States. Watts wondered if there was a connection between this apparent bit of folklore and the problem he was trying to solve and perhaps many others too.
Watts expected his idea would be ridiculed by his advisor, Steve Strogatz at Cornell's department of theoretical and applied mechanics. Instead, Stogatz also fell under the spell of the big mystery of the small world effect, and the two joined forces to try and solve it.
They began by using a computer to create lots of networks of virtual "friends", and measuring how many "handshakes" were needed to connect one friend to another in a totally different part of the network.
At one extreme were the utterly regular networks, where every friend only knows those right next to them. Devoid of any long range connections capable of linking, say, Bill Clinton to some store-keep in Hawaii, these networks typically demanded lots of handshakes before one person could be connected to another. Right at the other extreme were totally random networks, where people were just as likely to have personal friends in the White House as in Hawaiian stores.
Watts and Strogatz were intrigued by what happened between these two extremes, when the network was neither entirely regular nor utterly random. They expected the number of handshakes needed to link people to drop as the random links grew.
But what they discovered was startling: just a tiny number of random links was enough to "short-circuit" an otherwise huge, regular network, allowing apparently unrelated friends to be linked in just a few handshakes.
The computer revealed how easy it is to turn even a vast network into a small world: if only one in 100 people have a random link to anyone else in the network, the average number of handshakes drops ten-fold.
THEY ARE EVERYWHERE
But computer simulations are one thing; can small worlds be created so easily in the real one? Watts and Strogatz set about searching for a huge real-life network that they could probe for signs of the small world effect. They found the perfect, if unlikely, example in the Internet Movie Database, a computer searchable catalogue with the names of over 200,000 actors and the films they have appeared in.
Analyzing the database, Watts and Strogatz found that the typical actor has worked with around 60 others. If the showbiz network were completely regular, with no random short-circuits, that figure would imply that you'd typically have to go through 1,800 other actors and their films to link one actor to another. Yet the computer showed that it is possible to link any actor to any other via just three intermediaries. The vast movie business is really a small world.
In fact, this had been known for years by movie buffs who play the so-called Kevin Bacon Game. The aim of the game is to link the eponymous American actor to any other via the fewest number of intermediaries.
Players were often struck by how often they could answer with the names of very few actors. For example, Bacon can be linked to Charlie Chaplin in just three steps: Bacon played in a film with Laurence Fishburn, who in turn was in a film with Marlon Brando, who himself once appeared with Chaplin.
Watts and Strogatz had confirmed what many players suspected was the explanation: the "short-circuiting" effect of a handful of actors whose careers span different eras, genres and cultures. For example, by starring in both Lethal Weapon and Hamlet, Mel Gibson short-circuits the all-action and classical genres, while martial arts actor Bruce Lee links the Chinese film industry to Hollywood.
The world of showbiz is now recognized as a classical small world. That is, it is made up of lots of little cliques of actors, most of whom stay in their own patch of the industry, mixed in with a few highly versatile ones with random links right across the acting network who thus link every actor to every other via very few steps.
When Watts and Strogatz published their findings in the leading science journal Nature, it triggered an explosion of media coverage. But it also sparked interest among academics in a diverse range of fields, all wondering whether small world effects are at work in their own patches.
The corporate world does show signs of being under the influence of small world effects, according to Bruce Kogut of the Wharton School of Business at the University of Pennsylvania, and Gordon Walker at the Cox School of Business at the Southern Methodist University in Texas.
Kogut and Walker applied Watts and Strogatz's methods to the ownership networks spanning over 500 of Germany's biggest corporations. Predictably, they found lots of cliques in the ownership of firms, the result of various tie-ups and mergers. But they also found that the short-circuiting effect of a few corporations typically allowed the ownership of any one firm to be linked to any other via just four intermediaries. In other words, for all their diversity, Germany's biggest firms actually form a cosy small world.
This explains why firms with apparently tenuous links to one another can still show similar corporate behavior. But it may also have implications for the way these companies deal with globalization.
Small world theory shows it only takes a few random links to short-circuit a vast network. So it's likely that the whole corporate world has already become a small world.
The study of small worlds is still in its infancy, yet it is already clear that their presence holds both benefits and threats. Economists and business studies experts are likely to reveal many more examples of small worlds and their implications in the years ahead.
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