Six degrees of separation, a misread experiment

Stanley Milgram gave people in the American Midwest a package intended for a target in Massachusetts, with instructions to pass it to an acquaintance judged closer to the target. Only people known on a first-name basis could serve as relays.

Among the chains that completed, the median number of intermediaries is about five, hence the six degrees formula, which Milgram himself did not use. It took hold later, notably through John Guare's play Six Degrees of Separation, first staged in 1990.

The idea of a world in which everyone is linked to everyone else by a short chain was in fact older: the Hungarian writer Frigyes Karinthy had dramatised it in a 1929 short story. Milgram's experiment gave it the appearance of a measurement.

A minority of the chains launched reached their destination. Broken chains are not counted, which biases the median downwards: long chains have more opportunities to break.

That objection, formulated notably by Judith Kleinfeld in the 2000s from Milgram's archives, is the central methodological criticism and has never been entirely lifted. Kleinfeld also noted that the participants had not been drawn at random from the population.

Peter Dodds, Roby Muhamad and Duncan Watts published in 2003 an email replication covering tens of thousands of participants and targets spread across several countries.

The completion rate there is lower still, and the authors estimate, after correcting for attrition, a median length compatible with the announced order of magnitude. They also observe that drop-outs stem mainly from participants' lack of interest, not from the absence of a possible relay.

Direct measurement on the graphs of large social networks finds short average distances, of the order of four to five intermediaries. These measurements no longer depend on the goodwill of participants: they compute paths across the whole graph.

Those measures concern ties declared on a platform, which is not the same as ties that can be mobilised. A short path existing does not mean a person can find it, and Milgram's experiment measured precisely that ability to find one's way, not just the distance.

Milgram and his successors note that chains pass through a small number of highly connected intermediaries, which concentrates traffic on a few people.

It is that point, rather than the number six, which fed the later network literature, notably the small-world model proposed by Duncan Watts and Steven Strogatz in 1998 and work on the properties of graphs with highly heterogeneous degree.