Information about Small World Phenomenon

The small world experiment comprised several experiments conducted by Stanley Milgram to investigate the small world phenomenon by examining the average path length for social networks of people in the United States. The research was groundbreaking in that it revealed that human society is a small world type network characterized by shorter-than-expected path lengths. The experiments are often associated with the term six degrees of separation, although Milgram did not use this term himself.

Historical context of the small world problem

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. Technological advances in the early 20th century inspired a Hungarian author, Frigyes Karinthy, to write, among many things, 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.

Stanley Milgram re-visited this idea indirectly through a landmark set of experiments beginning in 1967 at Harvard University in Cambridge, Massachusetts, USA. Milgram was a renowned experimental social psychologist. Perhaps his most famous work is a study of obedience and authority, which is widely known as the Milgram Experiment. Milgram, however, like Karinthy, was also fascinated by the increasing interconnectedness among human beings. He thus sought to devise an experiment that could answer the small world problem. It is unclear whether Milgram was directly influenced by Karinthy's work, though it is remarkable how similar the two pieces are ^[1]

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 ^[2]. 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 Omaha, Wichita, and Boston, to be the start and end points 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" is 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 (see Fig. 2). 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 ^[1] (see Fig. 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 (See Fig. 2). And, although Milgram himself never used the term six degrees of separation, these findings likely contributed to its widespread acceptance ^[1].

In an experiment where 160 letters were mailed out, 24 reached the target in his Sharon, MA 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.^[3]

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 ^[2]. This suggests that participants strongly favored geographic characteristics when choosing an appropriate next person in the chain.

Critiques

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 (see Fig. 3), 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' recent book on the topic:
4. It is impossible for the entire human population to be acquainted within six degrees of separation because of the existence of certain populations which have had no contact with people outside their own culture, such as the Sentinelese people of North Sentinel Island. Even so, proof that people with even remote connections is available.

Influence

The social sciences

The Tipping Point by Malcolm Gladwell, based on articles originally published in The New Yorker, 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).

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 in New York, USA is currently conducting an email-based version of the same experiment, and has actually found average path lengths of about five on a worldwide scale. 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 manmade 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:

"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."^[4]

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 like Friendster, MySpace, Facebook, Orkut, Cyworld, Bebo, and others have greatly increased the connectivity of the online space through the application of social networking concepts.

Six degrees of Kevin Bacon is a popular game based upon the notion of six degrees of separation. The Oracle of Bacon 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 Erdos Number, a measure of one's closeness to the prolific mathematician, Paul Erdos.

See also

• Erdős number
• Bacon number
• Random network
• Small-world networks
• Social networks
• Scale-free networks

External links

Is it possible that anyone in the world could reach anyone else through a chain of just six friends? There are three projects now testing this hypothesis:

• Small World Project - Columbia University
• The Electronic Small World Project
• The Small World Experiment - 54 little boxes travelling the world

Gladwell's original New Yorker article:

• Six Degrees of Lois Weisberg

Could It Be a Big World After All?

• What the Milgram Papers in the Yale Archives Reveal About the Original Small World Study

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
• Collaboration distance among writers of mathematical papers — requires login {BROKEN LINK}
• 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.

References