Small-world experiment

One possible path of a message in the "Small World" experiment by Stanley Milgram.

1. 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.[3]
2. 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.
3. 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.
4. 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.
5. 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.

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.[5]

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.[3]

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.[6]

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.[5] This suggests that participants strongly favored geographic characteristics when choosing an appropriate next person in the chain.

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:

1. Judith Kleinfeld argues[7] 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.[8]
2. 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 the greedy algorithm that makes local decisions only.
3. 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.
4. Some communities, such as the Sentinelese, are completely isolated, disrupting the otherwise global chains. Once these people are discovered, they remain more "distant" from the vast majority of the world, as they have few economic, familial, or social contacts with the world at large; before they are discovered, they are not within any degree of separation from the rest of the population. However, these populations are invariably tiny, rendering them of low statistical significance.

In addition to these methodological criticisms, conceptual issues are debated. One regards the social relevance of indirect contact chains of different degrees of separation. Much formal and empirical work focuses on diffusion processes, but the literature on the small-world problem also often illustrates the relevance of the research using an example (similar to Milgram's experiment) of a targeted search in which a starting person tries to obtain some kind of resource (e.g., information) from a target person, using a number of intermediaries to reach that target person. However, there is little empirical research showing that indirect channels with a length of about six degrees of separation are actually used for such directed search, or that such search processes are more efficient compared to other means (e.g., finding information in a directory).[9]

The Tipping Point by Malcolm Gladwell, based on articles originally published in The New Yorker,[10] elaborates on the "funneling" concept. Gladwell condenses sociological research, which argues that the six-degrees phenomenon is dependent on a few extraordinary people ("connectors") with large networks of contacts and friends: these hubs then mediate the connections between the vast majority of otherwise weakly connected individuals.

Recent work in the effects of the small world phenomenon on disease transmission, however, have indicated that due to the strongly connected nature of social networks as a whole, removing these hubs from a population usually has little effect on the average path length through the graph (Barrett et al., 2005).

Smaller communities, such as mathematicians and actors, have been found to be densely connected by chains of personal or professional associations. Mathematicians have created the Erdős number to describe their distance from Paul Erdős based on shared publications. A similar exercise has been carried out for the actor Kevin Bacon and other actors who appeared in movies together with him — the latter effort informing the game "Six Degrees of Kevin Bacon". There is also the combined Erdős-Bacon number, for actor-mathematicians and mathematician-actors. Players of the popular Asian game Go describe their distance from the great player Honinbo Shusaku by counting their Shusaku number, which counts degrees of separation through the games the players have had.[11]

In 1998, Duncan J. Watts and Steven Strogatz from Cornell University published the first network model on the small-world phenomenon. They showed that networks from both the natural and man-made world, such as power grids and the neural network of C. elegans, exhibit the small-world phenomenon. Watts and Strogatz showed that, beginning with a regular lattice, the addition of a small number of random links reduces the diameter—the longest direct path between any two vertices in the network—from being very long to being very short. The research was originally inspired by Watts' efforts to understand the synchronization of cricket chirps, which show a high degree of coordination over long ranges as though the insects are being guided by an invisible conductor. The mathematical model which Watts and Strogatz developed to explain this phenomenon has since been applied in a wide range of different areas. In Watts' words:[13]

I think I've been contacted by someone from just about every field outside of English literature. I've had letters from mathematicians, physicists, biochemists, neurophysiologists, epidemiologists, economists, sociologists; from people in marketing, information systems, civil engineering, and from a business enterprise that uses the concept of the small world for networking purposes on the Internet.

Generally, their model demonstrated the truth in Mark Granovetter's observation that it is "the strength of weak ties" that holds together a social network. Although the specific model has since been generalized by Jon Kleinberg, it remains a canonical case study in the field of complex networks. In network theory, the idea presented in the small-world network model has been explored quite extensively. Indeed, several classic results in random graph theory show that even networks with no real topological structure exhibit the small-world phenomenon, which mathematically is expressed as the diameter of the network growing with the logarithm of the number of nodes (rather than proportional to the number of nodes, as in the case for a lattice). This result similarly maps onto networks with a power-law degree distribution, such as scale-free networks.

In computer science, the small-world phenomenon (although it is not typically called that) is used in the development of secure peer-to-peer protocols, novel routing algorithms for the Internet and ad hoc wireless networks, and search algorithms for communication networks of all kinds.