It started with a breakdown that crossed my feed: its author went back to Stanley Milgram’s primary sources and recalculated the famous “six degrees of separation.” There was no grand debunking flourish in it. But there was one detail that caught harder than anything else. In the very first experiment—the one that began this whole story—three of sixty envelopes reached their destination. The report on it gathers dust in an archive and was never published.
Which raises the question: who, exactly, measured the six? Plenty of people invented it, but whether anyone ever got that number as a result remains an open question. It appears in none of Milgram’s tables, and it has no confidence interval. There is a 1929 fiction, a 1967 experiment, and a 1990 play, and along that chain “six” is passed from hand to hand like a rumor. What was measured, though, was something else: how many letters failed to arrive, and who passed along the ones that did. That is where the more interesting story turns up.
In 1929, the Hungarian satirist Frigyes Karinthy published a collection called Everything Is Different, containing a short story, “Chain-Links.” A group of friends argue over whether the world is shrinking; one bets that any one of Earth’s 1.5 billion inhabitants can be reached through no more than five people, one of whom is a personal acquaintance.
Then comes the game. How do you get in touch with Selma Lagerlöf, who has just won the Nobel Prize in Literature? Easy: the award was presented by King Gustav; the King loves tennis and has played with Béla Kehrling; and Kehrling is personally known to the narrator. Two links out of five. Care for a harder problem?
He solves a harder one himself. A riveter on Ford’s assembly line can be reached in four steps: the worker knows a foreman, the foreman knows Ford himself, Ford is close with the director of the Hearst publishing house, and he knows a friend of the narrator’s. The whole idea is purely literary, and the story’s main plot is about a man who gets hooked on the game the way people get hooked on gambling: he begins laying out chains not just between people but between random objects, “like a gambler who has lost everything and is playing for dried beans.”
And here comes the first snag. The wording in the story permits two readings: it is unclear whether those “five people” include the person being sought. It may mean five links—or six. The authors of a 2011 paper on distances in the Facebook graph honestly noted this ambiguity and added that the phrase “six degrees of separation” caught on not because of Karinthy, but after a play that appeared sixty-one years later.
Milgram did not take up this problem from scratch. The mathematical model of the small world had been devised back in the 1950s by Ithiel de Sola Pool of MIT and Manfred Kochen of IBM, but for twenty years they did not dare publish it: they felt they had not “cracked” the problem. Milgram, at Harvard, decided he would crack it—and turned it into an experiment.
The first round involved residents of Wichita, Kansas, while the target was the wife of a theological-seminary student in Cambridge, Massachusetts. Participants received an envelope, and it was no ordinary envelope: heavy blue card stock, “Harvard University” embossed in gold, inside a list of signatures made with a pen in different inks. It looked like a real passport. Four days after the mailing, a professor at the Episcopal Theological Seminary walked right up to the target on the street: “Alice, this is for you.” At first she thought someone had returned an envelope that had gotten lost and never left Cambridge at all. The chain turned out to be: a wheat farmer → an Episcopal priest in his town → a seminary professor in Cambridge → the target. Two intermediaries, four days.
This was the episode Milgram published—with the caveat that it was one of the shortest chains he received at all. He printed the statistics from the first study nowhere: neither in the popular article nor in an academic one. Judith Kleinfeld, a professor at the University of Alaska, found them later—in Milgram’s archive at Yale Library. Boxes 48 and 49 of the Milgram archive contain an undated manuscript, “Results of Communication Project”: sixty starters were recruited through a newspaper ad, fifty chains were launched, three envelopes out of sixty reached the target—5 percent—via an average of eight intermediaries, meaning nine handshakes.
The second study used residents of Nebraska, and its target was a stockbroker who worked in Boston and lived in the town of Sharon. Here, the numbers exist. In the 1967 popular article: of 160 chains started in Nebraska, 44 arrived and 126 broke down. In Travers and Milgram’s 1969 technical report, the count is different because the base is different: 296 starters, 217 of whom sent the envelope on at all, and 64 who made it to the target—29 percent. The median was five intermediaries; the mean, 5.2. Boston chains arrived in 4.4 intermediaries, Nebraska chains in 5.7, and the difference was statistically significant: geography matters.
Far more interesting is how the letters arrived. All 64 envelopes that made it through passed through 26 people. Sixteen of them—a quarter of all successes—were handed to the broker by the same man: Jacobs, a clothing merchant from Sharon. Another ten passed through Jones, five through Brown; both worked at the brokerage.
48 percent of all successful chains were accounted for by three people. Milgram wrote that this was “a considerable surprise and even shock” to the broker: it turned out his connection to the world rested on a small-town shopkeeper and two colleagues. The budget for the entire undertaking, according to the archival papers, was $680.
Kleinfeld did not come to the archive to debunk anything. She wanted to replicate the experiment in the internet age, and for that she was looking for what, exactly, the envelope looked like. Instead, she found something else.
First, that very manuscript about three letters arriving out of sixty. Second, the ad used to recruit participants in Kansas: it was aimed at outgoing people proud of their connections, and bore little resemblance to a representative sample. Third, a budget line for purchasing mailing lists: people whose names are sold are, on average, wealthier and better connected. Both key studies recruited starters from people who plainly had more social capital than average.
Then came a systematic finding. The 1968 Beck and Kadamani study (never published) of 151 volunteers from Crestline, Ohio: success was below 18 percent, and not a single low-income sender was able to reach a target of a different income level. The 1970 Korte and Milgram paper on the racial boundary: of 270 chains sent by white starters to African American targets in New York, 13 percent got through; to white targets, 33 percent. Lin and colleagues in 1978: 596 packages, 375 forwarded, 112 delivered (30 percent), with the conclusion that communication almost never crosses the racial boundary. Kleinfeld specifically looked for studies where the experiment went beyond a single city: she found two such replications, one of them conducted by Milgram himself.
The sole exception was Gyo’s 1976 study in Montreal, with an 85 percent success rate, but first, they called on the phone rather than writing letters, and second, after a refusal they restarted the chain from the previous link. What this measured was the administrator’s patience more than the network’s properties.
So why does everyone believe in “six degrees of separation”? Kleinfeld asked people this question and got three recurring answers. A feeling of safety—“life is scary; it is nice to think we are all holding hands.” A religious interpretation: you meet someone from your past, so it was meant to happen.
The third was a complete failure to understand the mathematics of coincidence: one of her interviewees happily told her how he ran into a former classmate on a train in Belgium, only afterward admitting that he had also run into the same woman at an American Express office in Rome—and in Paris. There is another confusion she points out: “what is the probability that two random people have an acquaintance in common” is not the same as “what is the probability that at some point in your life you will run into anyone at all from your past.” The second probability is not small at all and requires no small world.
Kleinfeld puts her conclusion without academic politeness: judging by the evidence, we do not live in an elegant mathematical pattern where a random connector stitches everyone to everyone else, but in “a bowl of lumpy oatmeal.”
The phrase everyone quotes was born in the theater. In 1990, John Guare premiered his play Six Degrees of Separation: Off-Broadway in May, moved to Broadway in November, 485 performances, Pulitzer and Tony nominations. Ouisa, the owner of an apartment by Central Park, says: “I read somewhere that everybody on this planet is separated by only six other people... The President of the United States. A gondolier in Venice. Just fill in the names... But you have to find the right six people.”
Not a bad story in its own right: the play grew out of a criminal case. In October 1983, a young man named David Hampton appeared at the New York apartment of Inger and Osborne Elliott, claiming to be Sidney Poitier’s son. They let him stay the night, and in the morning the hostess found the guest in bed with another man and called the police. Acquaintances told Guare the story, and seven years later it became a play. Hampton tried to sue the playwright for a share of the profits as the real-life prototype—without success; he died in 2003.
After that, the number flowed downstream through culture. In 1994, three students at Albright College in Pennsylvania invented the game “Six Degrees of Kevin Bacon”—link any actor to Bacon through six films (legend has it that the idea came during a viewing of Footloose). In 1997, Andrew Weinreich launched SixDegrees.com—a social network named directly after the concept: it allowed users to write to people in their first, second, and third circles and see their connection to any user. By 1999 it had 3.5 million fully registered users, the company was bought for $125 million, and on December 30, 2000, the site shut down. One of the world’s first social networks bore the name of a number that nobody had yet measured.
Mathematicians picked up the theme later and from another angle. In 1998, Duncan Watts and Steven Strogatz showed in Nature that it was enough to add a few random long-range links to a regular lattice for distances within it to collapse sharply while high clustering remained. It also explained why the “small world” appears in both a worm’s neural network and the US power grid.
Then, in 2000, Jon Kleinberg separated two questions that had previously been fused together. First: do short paths exist in a network? Second: can anyone find them using only local information?
The answer to the second proved unpleasant. In the Watts–Strogatz model, short paths exist, but no algorithm can find them using only local links: expected delivery time grows as n to the two-thirds power. The only distribution of long-range links under which the greedy strategy—“send it to whoever is closest to the target”—works is inverse-square; in that case, delivery takes on the order of the logarithm squared. In a lattice of dimension k, navigation works if and only if the distribution’s exponent equals that very k. The subtlety is that this measure is not about the network’s diameter: short paths are abundant, but without clues about the geometry, they cannot be found.
Kleinberg, incidentally, cites Milgram as the original source of the intuition: “The geographical movement of the envelope from Nebraska to Massachusetts is striking—at each new person in the chain there is progressive convergence upon the target area.”
This was confirmed experimentally in 2003. Peter Dodds, Roby Muhamad, and Watts himself opened a website where 98,847 people registered; about a quarter of them went on to launch a chain. In the end, they had data on 61,168 participants from 166 countries and 24,163 individual chains. Just 384 reached their targets—about 1.6 percent.
Dropout at every subsequent step was around 63 percent; at the first, a full 75 percent. Fewer than 0.3 percent of those who dropped out said they had not found a suitable recipient: the rest simply did not bother. Recalculating the distribution for “ideal” conditions without dropout, the authors got a median of 5 steps for pairs within one country and 7 for pairs in different countries. This is the most honest version of “six degrees of separation”: a model estimate adjusted for the fact that people quit the game.
Then came the big graphs. In 2008, Leskovec and Horvitz processed one month of Microsoft Messenger logs: 30 billion conversations, 240 million people, a graph of 180 million nodes and 1.3 billion links, with an average path length of 6.6. In 2011, Facebook calculated the average distance across 721 million users and 69 billion friendships: 4.74. In 2016, with 1.59 billion users: 4.57.
This is where arithmetic confusion begins, neatly demonstrating how “facts” are born. That same 4.74 is a distance—that is, 3.74 intermediaries. The press and Wikipedia tables cite 4.74, while Facebook’s blog and the paper itself say 3.74 degrees; for 2016, the same table gives 4.57, although the blog itself gives 3.57. Two counting conventions live in one column. A similar discrepancy surrounds the Colombian email experiment: the academic paper says 18 targets in 13 countries; Wikipedia says 19 targets in 157 countries. Both figures travel the world as equals.
And here the picture flips. All these large numbers are about the graph, not navigation. In 2014, a Hungarian group—symbolically enough, from Karinthy’s country—ran Kleinberg’s greedy routing across a graph of geolocated tweets: 5.8 million nodes, 122 million edges, 4 million routes. The average actual distance between users was 4.95 steps. A message reaches the target’s area in 3–6 steps. But it reaches a specific addressee in roughly 20 percent of cases. One of the examples in the paper: a route from Wichita to Boston.
The algorithm crosses half the country in a few steps, then spends 90 percent of the time circling the target’s vicinity, unable to find the final step. Milgram explained his 29 percent by saying that people were lazy; the Hungarians offer a second explanation—the network simply cannot be navigated at city-scale distances.
The number six was not measured in a single study.
It has a literary grandfather—a satirist’s invention, whose author did not himself know whether he meant five links or six—and a theatrical father, a playwright who attributed the idea to Marconi. Everything actually measured in the experiments concerns other quantities: how quickly chains fall apart, who ends up at the penultimate link, and how social barriers—class and race—cut the network into pieces. The accurate wording about path length is this: 5–7 steps in a model adjusted for dropout, and 4–6 in large graphs, while no one inside the network can actually traverse the path.
The distinction invented by Kleinberg is the answer to the original question. The world is small as a graph and almost unnavigable as a route. Six degrees of separation are not about how far away the person you need is; they are about the fact that distance in a graph and a road you can actually travel are different things. The number everyone knows describes the first. We live in the second.
One detail remains—the one that started it all. In box 48 of Yale Library lies a manuscript in which three of sixty envelopes arrived. Milgram did not publish it. But he did publish the anecdote about four days.