One day during the winter of 1998, mathematicians Duncan Watts and Steve Strogatz of Cornell University in Ithaca, New York, sat down at a table in Strogatz’s office and drew a series of dots on a piece of paper. They then connected some of the dots together with lines to produce a simple pattern that mathematicians refer to as a graph. This may not sound like serious mathematics; it certainly does not sound like a profitable way to go about making discoveries. But as the two mathematicians were soon to learn, they had connected their dots in a peculiar way that no mathematician had ever envisioned. In so doing, they had stumbled over a graph of an unprecedented and fascinating kind.
Watts and Strogatz came upon their graph while trying to make sense of a curious puzzle of our social world. In the 1960s, an American psychologist named Stanley Milgram tried to form a picture of the web of interpersonal connections that link people into a community. To do so, he sent letters to a random selection of people living in Nebraska and Kansas, asking each of them to forward the letter to a stockbroker friend of his living in Boston, but he did not give them the address. To forward the letter, he asked them to send it only to someone they knew personally and whom they thought might be socially “closer” to the stockbroker. Most of the letters eventually made it to his friend in Boston. Far more startling, however, was how quickly they did so—not in hundreds of mailings but typically in just six or so. The result seems incredible, as there are hundreds of millions of people in the United States, and both Nebraska and Kansas would seem a rather long way away—in the social universe—from Boston. Milgram’s findings became famous and passed into popular folklore in the phrase “six degrees of separation.” As the writer John Guare expressed the idea in a recent play of the same name:
“Everybody on this planet is separated by only six other people…. The president of the United States. A gondolier in Venice…. It’s not just the big names. It’s anyone. A native in the rain forest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people. It’s a profound thought….”
It is a profound thought, and yet it really seems to be true. A few years ago a German newspaper accepted the light-hearted challenge of trying to connect a Turkish kebab-shop owner in Frankfurt to his favorite actor, Marlon Brando. After several months, the staff of Die Ziet discovered that it took no more than six links of personal acquaintance to do so. The kebab- shop owner, an Iraqi immigrant named Salah Ben Ghaln, has a friend living in California. As it happens, this friend works alongside the boyfriend of a woman who is the sorority sister of the daughter of the producer of the film Don Juan de Marco, in which Brando starred. Six-degrees of separation is an undeniably stunning characteristic of our social world, and numerous more careful sociological studies offer convincing evidence that it is true— not only in special cases, but generally. But how can it be true? How can six billion people be so closely linked?
These are the questions that Watts and Strogatz set for themselves. If you think of people as dots, and links of acquaintance as lines connecting them, then the social world becomes a graph. So for months Watts and Strogatz had been drawing graphs of all sorts, connecting dots in different patterns, hoping to find some remarkable scheme that would reveal how six billion people could conceivably be connected together so closely. They tried drawing dots and linking them together into a grid, arranged regularly like the squares on a chessboard. They tried drawing dots and connecting them haphazardly to produce random graphs, each looking like a connect- the-dots game gone haywire. But neither the ordered nor the random graphs seemed to capture the nuances of real social networks. The small-world mystery remained defiant.
Then finally, on that winter’s day in 1998, the two researchers stumbled over their peculiar graph. What they discovered was a subtle way of connecting the dots that was neither orderly nor random but somewhere in- between, an unusual pattern with chaos mingling in equal balance with order. Playing with variations of this odd-looking graph over the next few weeks, Watts and Strogatz found that it held the key to revealing how it is that six billion people can be connected by only six links.
Small Worlds by Manuel Fraga is licensed under a Creative Commons Attribution 4.0 International License.