The small-world experiment comprised several experiments conducted by Stanley Milgram and other researchers examining the average path length for social networks of people in the United States. The research was groundbreaking in that it suggested that human society is a small-world-type network characterized by short path-lengths. The experiments are often associated with the phrase "six degrees of separation", although Milgram did not use this term himself.
Historical context of the small-world problem
Guglielmo Marconi's conjectures based on his radio work in the early 20th century, which were articulated in his 1909 Nobel Prize address,Guglielmo Marconi, 1909, Nobel Lecture, Wireless telegraphic communication. may have inspired Hungarian author Frigyes Karinthy to write a challenge to find another person to whom he could not be connected through at most five people.Barabási, Albert-László . 2003. "Linked: How Everything is Connected to Everything Else and What It Means for Business, Science, and Everyday Life. " New York: Plume. This is perhaps the earliest reference to the concept of six degrees of separation, and the search for an answer to the small world problem.
Mathematician Manfred Kochen and political scientist Ithiel de Sola Pool wrote a mathematical manuscript, "Contacts and Influences", while working at the University of Paris in the early 1950s, during a time when Milgram visited and collaborated in their research. Their unpublished manuscript circulated among academics for over 20 years before publication in 1978. It formally articulated the mechanics of social networks, and explored the mathematical consequences of these (including the degree of connectedness). The manuscript left many significant questions about networks unresolved, and one of these was the number of degrees of separation in actual social networks.
Milgram took up the challenge on his return from Paris, leading to the experiments reported in "The Small World Problem" in the May 1967 (charter) issue of the popular magazine Psychology Today, with a more rigorous version of the paper appearing in Sociometry two years later. The Psychology Today article generated enormous publicity for the experiments, which are well known today, long after much of the formative work has been forgotten.
Milgram's experiment was conceived in an era when a number of independent threads were converging on the idea that the world is becoming increasingly interconnected. Michael Gurevich had conducted seminal work in his empirical study of the structure of social networks in his MIT doctoral dissertation under Pool. Mathematician Manfred Kochen, an Austrian who had been involved in statist urban design, extrapolated these empirical results in a mathematical manuscript, Contacts and Influences, concluding that, in an American-sized population without social structure, "it is practically certain that any two individuals can contact one another by means of at least two intermediaries. In a [socially] structured population it is less likely but still seems probable. And perhaps for the whole world's population, probably only one more bridging individual should be needed." They subsequently constructed Monte Carlo simulations based on Gurevich's data, which recognized that both weak and strong acquaintance links are needed to model social structure. The simulations, running on the slower computers of 1973, were limited, but still were able to predict that a more realistic three degrees of separation existed across the U.S. population, a value that foreshadowed the findings of Milgram.
Milgram revisited Gurevich's experiments in acquaintanceship networks when he conducted a highly publicized set of experiments beginning in 1967 at Harvard University. One of Milgram's most famous works is a study of obedience and authority, which is widely known as the Milgram Experiment. Milgram's earlier association with Pool and Kochen was the likely source of his interest in the increasing interconnectedness among human beings. Gurevich's interviews served as a basis for his small world experiments.
Milgram sought to develop an experiment that could answer the small world problem. This was the same phenomenon articulated by the writer Frigyes Karinthy in the 1920s while documenting a widely circulated belief in Budapest that individuals were separated by six degrees of social contact. This observation, in turn, was loosely based on the seminal demographic work of the Statists who were so influential in the design of Eastern European cities during that period. Mathematician Benoit Mandelbrot, born in Poland and having traveled extensively in Eastern Europe, was aware of the Statist rules of thumb, and was also a colleague of Pool, Kochen and Milgram at the University of Paris during the early 1950s (Kochen brought Mandelbrot to work at the Institute for Advanced Study and later IBM in the U.S.). This circle of researchers was fascinated by the interconnectedness and "social capital" of social networks.
Milgram's study results showed that people in the United States seemed to be connected by approximately three friendship links, on average, without speculating on global linkages; he never actually used the phrase "six degrees of separation". Since the Psychology Today article gave the experiments wide publicity, Milgram, Kochen, and Karinthy all had been incorrectly attributed as the origin of the notion of "six degrees"; the most likely popularizer of the phrase "six degrees of separation" is John Guare, who attributed the value "six" to Marconi.
The experiment
Milgram's experiment developed out of a desire to learn more about the probability that two randomly selected people would know each other. This is one way of looking at the small world problem. An alternative view of the problem is to imagine the population as a social network and attempt to find the average path length between any two nodes. Milgram's experiment was designed to measure these path lengths by developing a procedure to count the number of ties between any two people.
Basic procedure
Though the experiment went through several variations, Milgram typically chose individuals in the U.S. cities of Omaha, Nebraska, and Wichita, Kansas, to be the starting points and Boston, Massachusetts, to be the end point of a chain of correspondence. These cities were selected because they were thought to represent a great distance in the United States, both socially and geographically. Information packets were initially sent to "randomly" selected individuals in Omaha or Wichita. They included letters, which detailed the study's purpose, and basic information about a target contact person in Boston. It additionally contained a roster on which they could write their own name, as well as business reply cards that were pre-addressed to Harvard. Upon receiving the invitation to participate, the recipient was asked whether he or she personally knew the contact person described in the letter. If so, the person was to forward the letter directly to that person. For the purposes of this study, knowing someone "personally" was defined as knowing them on a first-name basis. In the more likely case that the person did not personally know the target, then the person was to think of a friend or relative who was more likely to know the target. They were then directed to sign their name on the roster and forward the packet to that person. A postcard was also mailed to the researchers at Harvard so that they could track the chain's progression toward the target. When and if the package eventually reached the contact person in Boston, the researchers could examine the roster to count the number of times it had been forwarded from person to person. Additionally, for packages that never reached the destination, the incoming postcards helped identify the break point in the chain.
Results
Shortly after the experiments began, letters would begin arriving to the targets and the researchers would receive postcards from the respondents. Sometimes the packet would arrive to the target in as few as one or two hops, while some chains were composed of as many as nine or ten links. However, a significant problem was that often people refused to pass the letter forward, and thus the chain never reached its destination. In one case, 232 of the 296 letters never reached the destination.
However, 64 of the letters eventually did reach the target contact. Among these chains, the average path length fell around five and a half or six. Hence, the researchers concluded that people in the United States are separated by about six people on average. Although Milgram himself never used the phrase "six degrees of separation", these findings are likely to have contributed to its widespread acceptance.
In an experiment in which 160 letters were mailed out, 24 reached the target in his home in Sharon, Massachusetts. Of those 24 letters, 16 were given to the target by the same person, a clothing merchant Milgram called "Mr. Jacobs". Of those that reached the target at his office, more than half came from two other men.
The researchers used the postcards to qualitatively examine the types of chains that are created. Generally, the package quickly reached a close geographic proximity, but would circle the target almost randomly until it found the target's inner circle of friends. This suggests that participants strongly favored geographic characteristics when choosing an appropriate next person in the chain.
Criticisms There are a number of methodological criticisms of the small-world experiment, which suggest that the average path length might actually be smaller or larger than Milgram expected. Four such criticisms are summarized here:
Judith Kleinfeld argues that Milgram's study suffers from selection and non-response bias due to the way participants were recruited and high non-completion rates. First, the "starters" were not chosen at random, as they were recruited through an advertisement that specifically sought people who considered themselves well-connected. Another problem has to do with the attrition rate. If one assumes a constant portion of non-response for each person in the chain, longer chains will be under-represented because it is more likely that they will encounter an unwilling participant. Hence, Milgram's experiment should underestimate the true average path length. Several methods have been suggested to correct these estimates; one uses a variant of survival analysis in order to account for the length information of interrupted chains, and thus reduce the bias in the estimation of average degrees of separation. One of the key features of Milgram's methodology is that participants are asked to choose the person they know who is most likely to know the target individual. But in many cases, the participant may be unsure which of their friends is the most likely to know the target. Thus, since the participants of the Milgram experiment do not have a topological map of the social network, they might actually be sending the package further away from the target rather than sending it along the shortest path. This is very likely to increase route length, overestimating the average number of ties needed to connect two random people. An omniscient path-planner, having access to the complete social graph of the country, would be able to choose a shortest path that is, in general, shorter than the path produced by a greedy algorithm that makes local decisions only. A description of heterogeneous social networks still remains an open question. Though much research was not done for a number of years, in 1998 Duncan Watts and Steven Strogatz published a breakthrough paper in the journal Nature. Mark Buchanan said, "Their paper touched off a storm of further work across many fields of science" (Nexus, p60, 2002). See Watts' book on the topic: Six Degrees: The Science of a Connected Age.