The Small world phenomenon reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Small world phenomenon

The small world phenomenon is the theory that everyone in the world can be reached through a short chain of social acquaintances. This concept gave rise to the famous phrase six degrees of separation, implying that the chain is typically no more than six links long. However, after more than thirty years its status as a description of heterogeneous social networks (such as the aforementioned "everyone in the world") still remains an open question. Remarkably little research has been done in this area since the publication of the original paper.

Table of contents

1 1967 paper 2 Mathematicians and actors 3 Milgram's experiment 4 Results 5 Influence 5.1 The social sciences 5.2 Mathematics and other disciplines 6 Play/film 7 See also 8 Reference 9 External links

1967 paper

The idea is due to Stanley Milgram, and was first published in the popular magazine Psychology Today as "The Small World Problem" in 1967. A "technical report" was published in 1969 which filled in some of the details missing from the original paper.

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, and a similar exercise has been carried out for the actor Kevin Bacon - the latter effort informing the game "Six Degrees of Kevin Bacon".

Milgram's experiment

Milgram's original research - conducted among the population at large, rather than the specialized, highly collaborative fields of mathematics and acting - has been challenged on a number of fronts. In his first "small world" experiment (documented in an undated paper entitled "Results of Communication Project"), Milgram sent 60 letters to various recruits in Wichita, Kansas who were asked to forward the letter to the wife of a divinity student living at a specified location in Cambridge, Massachusetts. The particpants could only pass the letters (by hand) to personal acquaintances who they thought might be able to reach the target - whether directly or via a "friend of a friend". While fifty people responded to the challenge, only three letters eventually reached their destination. Milgram's celebrated 1967 paper refers to the fact that one of the letters in this initial experiment reached the recipient in just four days, but neglects to mention the fact that only 5% of the letters successfully "connected" to their target. In two subsequent experiments, chain completion was so low that the results were never published. On top of this, researchers have shown that a number of subtle factors can have a profound effect on the results of "small world" experiments. Studies that attempted to connect people of differing races or incomes showed significant asymmetries. Indeed a paper which revealed a completion rate of 13% for black targets and 33% for white targets (despite the fact that the participants did not know the race of the recipient) was co-written by Milgram himself.

Results

Despite these complications, a variety of novel discoveries did emerge from Milgram's research. After numerous refinements of the apparatus (the perceived value of the letter or parcel was a key factor in whether people were motivated to pass it on or not), Milgram was able to achieve completion rates of 35%, and later researchers pushed this as high as 97%. If there was some doubt as to whether the "whole world" was a small world, there was very little doubt that there were a large number of small worlds within that whole (from faculty chains at Michigan State University to a close-knit Jewish community in Montreal). For those chains that did reach completion the number 6 emerged as the mean number of intermediaries and thus the expression "six degrees of separation" (perhaps by analogy to "six degrees of freedom") was born. In addition, Milgram identified a "funneling" effect whereby most of the forwarding (i.e. connecting) was being done by a very small number of "stars" with significantly higher-than-average connectivity: even on the 5% "pilot" study, Milgram noted that "two of the three completed chains went through the same people".

Influence

The social sciences

The Tipping Point by Malcom Gladwell, based on articles originally published in The New Yorker, elaborates the "funneling" concept. In it 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.

Mathematics and other disciplines

In a paper published in the June 4 1998 edition of Nature, Duncan J. Watts and Steven H. Strogatz, then mathematicians at Cornell University, caused a stir by announcing that small world networks are common in a variety of different realms ranging from C. elegans neurons to power grids.

Watts and Strogatz show that the addition of a handful of random links can turn a disconnected network into a highly connected one. This has both positive and negative implications: it's a virtue if, by the addition of a few judicious routers, it makes a vast communication network (such as the Internet) no more than six hops wide; in contrast, it's a vice if it places that same well-connected individual a mere six people away from a deadly disease such as SARS.

The research was 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 Watts, in conjunction with his supervisor, 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." [1]

Play/film

Six Degrees of Separation is also the title of a play and film written by John Guare, based on the true story of a conman who bluffed his way into Manhattan high society by claiming to be the son of a famous actor.

See also

• Erdős number
• Bacon number
• Duncan J. Watts
• Random network
• Small-world networks
• Social networks

Reference

• Stanley Milgram, "The Small World Problem", Psychology Today, May 1967. pp 60 - 67.
• Malcolm Gladwell, The Tipping Point, 2000.
• Duncan J. Watts, Six Degrees: The Science Of A Connected Age, 2003.

External links

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

• Small World Project - Columbia University
• The Electronic Small World Project

About the play:

• Six Degrees of Separation at CU-Boulder

About small world networks:

• Small-world Networks

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

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