Small worlds and six degrees: what the folders, the model and the emails showed

The phrase is older than any measurement of it. In Frigyes Karinthy's 1929 short story "Chains", characters wager that anyone on earth can be reached through at most five acquaintances, and John Guare's 1990 play Six Degrees of Separation fixed the number six in everyday speech. In between came a social psychologist with a stack of folders, and later two mathematicians with a ring of dots. This explainer takes the evidence in order, then asks what it means for the smaller worlds anthropologists study: villages, lineages and the kin ties that join them.

The folders

Stanley Milgram described his method in the first issue of Psychology Today (1967), and he and Jeffrey Travers gave the full technical report in Sociometry (1969). Each starter received a folder naming a target, a stockbroker who lived in Sharon, Massachusetts, and worked in Boston. Starters learned his name, address, occupation, employer, college, years of military service, and his wife's maiden name and hometown. If they knew him personally, on a first-name basis, they could mail the folder straight to him. Otherwise they sent it to a first-name acquaintance more likely to know him, who did the same. Postcards returned at each step let the researchers follow every chain.

There were 296 starters: 100 Nebraskans who owned blue-chip stock, 96 other Nebraskans the paper labels "random", and 100 Boston-area volunteers recruited through a newspaper advertisement. Of these, 217 actually sent a folder on.

About five intermediaries means about six links from starter to target, which is where the folk number found its evidence.

The fine print

Most chains never arrived: 153 of the 217 died along the way, and 59 of them, more than a quarter of all chains, ended at the very first handoff. Travers and Milgram found no significant differences in age, sex or relationship between people who dropped out and people who carried on, so they treated dropouts as random. A footnote reports Harrison White's correction for those losses, under which the median length of completed chains rises from 5 to 7.

Arrivals also funneled. The 64 folders that reached the target came through only 26 people; a clothing merchant in Sharon delivered 16 of them, and he and two business associates of the target together delivered 48 percent. Senders mostly picked people like themselves in age, and men sent the folder to other men far more often than to women. Only 14 percent of handoffs went to relatives; the rest went to friends and acquaintances.

Decades later Judith Kleinfeld (2002, Society) went through Milgram's papers at Yale. She found an unpublished first study in which 60 starters, recruited in Wichita, Kansas, through a newspaper advertisement, tried to reach the wife of a divinity student in Cambridge, Massachusetts. Three of the 60 folders arrived. She also pointed out that the "random" Nebraskans came from mailing lists, and that in a later study by Charles Korte and Milgram, 13 percent of chains from white starters reached Black targets, against 33 percent reaching white targets. Her conclusion was that "six degrees" rested on thin evidence, and that barriers of race and class deserve as much attention as shortcuts.

The ring and the shortcuts

Duncan Watts and Steven Strogatz (1998, Nature) asked a structural question. A regular lattice, where everyone is tied to their neighbors, is clustered (your contacts know each other) but has long paths. A random network has short paths but almost no clustering. Real social networks seem to have both. Watts and Strogatz started from a ring lattice and rewired a small fraction of ties at random. Path length fell fast while clustering hardly moved, and they called the in-between networks small worlds. They showed that the collaboration graph of film actors, the power grid of the western United States and the neural network of the worm C. elegans all have this shape.

The two measures, for a network whose ties run both ways:

• Path length L: the fewest steps between two nodes, averaged over all pairs.
• Clustering C: for a node with k neighbors, the share of the k(k - 1)/2 possible ties among those neighbors that actually exist, averaged over all nodes.

A ring of 20 houses, by hand

Imagine 20 houses along a valley, numbered 0 to 19 and closed into a ring, each tied to the two nearest houses on either side, so 4 ties each. Think of the ties as kin and marriage links between neighbors.

Clustering for house 0: its neighbors are 1, 2, 18 and 19. Among four neighbors there are 4 × 3 / 2 = 6 possible ties. Three exist: 1 to 2, 18 to 19, and 19 to 1 (they are two apart, across house 0). So C = 3/6 = 0.5, and every house looks the same.

Path length for house 0: houses 1, 2, 18, 19 are one step away; 3, 4, 16, 17 two; 5, 6, 14, 15 three; 7, 8, 12, 13 four; and 9, 10, 11 five. The total is 4 × 1 + 4 × 2 + 4 × 3 + 4 × 4 + 3 × 5 = 55, so the average is 55 / 19 = 2.89. By symmetry that is L for the whole ring.

Now rewire one tie: drop the link from 0 to 2 and connect house 0 to house 10, on the far side. Think of a marriage that sends someone across the valley.

The new total is 4 × 1 + 8 × 2 + 7 × 3 = 41. House 0's clustering fell to 2/6 = 0.33, since only 1 to 19 and 18 to 19 now link its neighbors. Averaged over the whole network (190 pairs, quick by computer), the pattern looks like this, with a second rewiring that moves the tie from 5 to 7 so it runs from 5 to 15:

Two shortcuts cut the average path by about a sixth and clustering by less than a tenth. In a ring this small the effect is modest; in a long ring each shortcut spans a greater distance and the drop in L is far steeper, which is the heart of the Watts and Strogatz result.

Paths that exist, paths that can be found

Jon Kleinberg (2000, Nature) noticed that Milgram's study showed two things: short chains exist, and people with only local knowledge can find them. His model added long-range ties to a grid, and only one way of placing them let a simple forward-to-whoever-is-closest rule find short paths: long ties whose probability falls with the square of distance on a two-dimensional grid. In plain words, shortcuts have to come at every scale, many to the next village, fewer to the next region, fewer still to the far side of the world.

Peter Dodds, Roby Muhamad and Watts (2003, Science) repeated the search by email. Of 98,847 people who registered, about a quarter started chains aimed at one of 18 targets in 13 countries, including an archival inspector in Estonia and a policeman in Australia. Of 24,163 chains, 384 arrived, with an average length of 4.05. Correcting for attrition, they estimated a median of five to seven steps, depending on how far apart source and target were. Fewer than 0.3 percent of people who stopped said they could not think of anyone to send it to; chains died mostly from lack of interest. Successful chains leaned on professional ties (33.9 percent of links against 13.2 percent in failed chains) and rarely went through people with many friends.

The last row is a different kind of number. Lars Backstrom and colleagues computed distances across the whole Facebook graph of about 721 million users; nobody had to find those paths, and a Facebook friendship is a cheap tie.

Villages, kin and six degrees: four questions

Does six degrees mean everyone is closely related?

No. A chain of acquaintances is not a genealogy. Shared ancestry follows its own arithmetic, explained in our piece on pedigree collapse.

Is a village a small world?

Usually it is simply small, and whether short paths exist is the least interesting question about it. Even the 20-house ring above, with no shortcuts at all, puts every house within five steps of every other. What matters is clustering and the ties that bridge clusters. In kin-type shorthand (M mother, B brother, Z sister, H husband, W wife; ZH is sister's husband), a ZH or WB from another valley plays exactly the role of the rewired tie above, putting a whole distant household within two steps. Our explainer on Dunbar's number looks at how large personal networks can be.

Does a short path mean help or news will travel?

Not by itself. Travers and Milgram lost 71 percent of their chains, and Dodds and colleagues lost far more, mostly to indifference. Kleinfeld's racial gap shows that some short paths exist on paper and are rarely used. In kin networks the kind of tie matters as much as its length, because obligations to a B, an MB or a WB are not the same.

How can I try it?

List the ties of a small network, then count steps outward from one node, as in the ring above. Our first network measures explainer and the network density calculator cover the counting. A starter exercise: 12 houses in a ring, each tied to two neighbors on each side. House 0's average distance is 21 / 11 = 1.91. Add one marriage tie from house 0 to house 6, keeping all the old ties, and it drops to 17 / 11 = 1.55.

Sources and further reading

• Milgram, Stanley. 1967. "The small-world problem." Psychology Today 1 (1), May 1967.
• Travers, Jeffrey, and Stanley Milgram. 1969. "An experimental study of the small world problem." Sociometry 32 (4): 425-443.
• Kleinfeld, Judith S. 2002. "The small world problem." Society 39 (2): 61-66.
• Watts, Duncan J., and Steven H. Strogatz. 1998. "Collective dynamics of 'small-world' networks." Nature 393: 440-442. https://doi.org/10.1038/30918
• Kleinberg, Jon M. 2000. "Navigation in a small world." Nature 406: 845. https://doi.org/10.1038/35022643
• Dodds, Peter Sheridan, Roby Muhamad, and Duncan J. Watts. 2003. "An experimental study of search in global social networks." Science 301: 827-829. https://doi.org/10.1126/science.1081058
• Backstrom, Lars, Paolo Boldi, Marco Rosa, Johan Ugander, and Sebastiano Vigna. 2012. "Four degrees of separation." Proceedings of the 4th Annual ACM Web Science Conference: 33-42. https://arxiv.org/abs/1111.4570