Small world phenomenon.html



The small world experiment comprised several experiments conducted by Stanley Milgram 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, may have inspired^{citation needed} Hungarian author Frigyes Karinthy to write a challenge to find another person through which he could not be connected to by at most five people.¹ 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 Their unpublished manuscript circulated among academics for over 20 years before publication in 1978.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 primitive 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 work is a study of obedience and authority, which is widely known as the Milgram Experiment.² 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 devise 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.³ 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

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 represented a great distance in the United States, both socially and geographically.¹
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 they know personally that is 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.

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.³

However, 64 of the letters eventually did reach the target contact. Among these chains, the average path length fell around 5.5 or six. Hence, the researchers concluded that people in the United States are separated by about six people on average. And, although Milgram himself never used the phrase "six degrees of separation", these findings likely contributed to its widespread acceptance.¹

In an experiment in which 160 letters were mailed out, 24 reached the target in his Sharon, Massachusetts home. Of those 24, 16 were given to the target person by the same person Milgram calls "Mr. Jacobs", a clothing merchant. Of those that reached him at his office, more than half came from two other men.⁴

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.³ This suggests that participants strongly favored geographic characteristics when choosing an appropriate next person in the chain.

Critiques

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There are a number of methodological critiques of the Milgram Experiment, which suggest that the average path length might actually be smaller or larger than Milgram expected. Four such critiques are summarized here:

1. The "Six Degrees of Separation" Myth argues that Milgram's study suffers from selection and nonresponse bias due to the way participants were recruited and high non-completion rates. 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 under-estimate the true average path length.
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 may create a slight bias and over-estimate the average number of ties needed for two random people.
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 are completely isolated, such as the Sentinelese of North Sentinel Island, disrupting the otherwise global chains. Once these people are discovered, they by necessity 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 – the Sentinelese people number 250 – rendering them, arguably, of low statistical significance to the overall result.

Influence

The social sciences

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The Tipping Point by Malcolm Gladwell, based on articles originally published in The New Yorker,⁵ elaborates the "funneling" concept. Gladwell 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).^{citation needed}

Mathematicians and actors

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 for actors who appeared in movies together — 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.

Current research on the small world problem

The small world question is still a popular research topic today, with many experiments still being conducted. For instance, the Small World Project at Columbia University conducted an email-based version of the same experiment, and found average path lengths of about five on a worldwide scale. A similar experiment using popular social networking sites as a median was carried out at Carnegie Mellon University. Results showed that very few messages actually reached their destination. However, the critiques that apply to Milgram's experiment largely apply also to this current research.

Network models

In 1998, Duncan J. Watts and Steven H. Strogatz, both in the Department of Theoretical and Applied Mechanics at 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 the neural network of C. elegans and power grids, exhibit the small-world property. 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:⁶

"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.

Milgram's experiment in popular culture

Social networks pervade popular culture in the United States and elsewhere. In particular, the notion of six degrees has become part of the collective consciousness. Social networking websites such as Friendster, MySpace, XING, Orkut, Cyworld, Bebo, Facebook, and others have greatly increased the connectivity of the online space through the application of social networking concepts. The "Six Degrees" Facebook application⁷ calculates the number of steps between any two members.

"Six Degrees of Kevin Bacon" is a game based upon the notion of six degrees of separation. The website The Oracle of Bacon⁸ uses social network data available from the Internet Movie Database to determine the number of links between Kevin Bacon and any other celebrity. One academic variant of the game involves calculating an Erdős number, a measure of one's closeness to the prolific mathematician, Paul Erdős.

See also

• Bacon number
• Random network
• Small-world network

References

External links

• A new Small World Problem Experiment
• The Small World Experiment - 54 little boxes travelling the world
• Find Satoshi Experiment
• What the Milgram Papers in the Yale Archives Reveal About the Original Small World Study
• Planetary-Scale Views on an Instant-Messaging Network
• Mission of the world - Stuffed animals travelling the world on different missions
• Collective dynamics of small-world networks:
  • Explaining the "Small World" Phenomenon
• Theory tested for specific groups:
  • The Oracle of Bacon at Virginia
  • The Oracle of Baseball
  • The Erdős Number Project
  • The Oracle of Music
  • CoverTrek - linking bands and musicians via cover versions.
  • Science Friday: Future of Hubble / Small World Networks
  • Knock, Knock, Knocking on Newton's DoorPDF (223 KiB) - article published in Defense Acquisition University's journal Defense AT&L, proposes "small world / large tent" social networking model.
  • The Chess Oracle of Kasparov - the theory tested for chess players.