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The small world phenomenon (also known as the small world effect) is the hypothesis that everyone in the world can be reached through a short chain of social acquaintances. The concept gave rise to the famous phrase six degrees of separation after a 1967 small world experiment by social psychologist Stanley Milgram which suggested that two random US citizens were connected on average by a chain of six acquaintances. Image File history File links Unbalanced_scales. ... A hypothesis (from Greek ) is a suggested explanation of a phenomenon or reasoned proposal suggesting a possible correlation between multiple phenomena. ... This article or section is in need of attention from an expert on the subject. ... 1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar (the link is to a full 1967 calendar). ... A psychologist is a scientist and/or clinician who studies psychology, the systematic investigation of the human mind, including behavior and cognition. ... Stanley Milgram Stanley Milgram (August 15, 1933 – December 20, 1984) was a psychologist at Yale University, Harvard University and the City University of New York. ... Motto: (Out Of Many, One) (traditional) In God We Trust (1956 to date) Anthem: The Star-Spangled Banner Capital Washington D.C. Largest city New York City None at federal level (English de facto) Government Federal constitutional republic  - President George Walker Bush (R)  - Vice President Dick Cheney (R) Independence from...

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. Little research has been done in this area since the publication of the original paper. A social network is a social structure made of nodes which are generally individuals or organizations. ...

Milgram's experiment

Milgram's original research — conducted among the population at large, rather than the specialized, highly collaborative fields of mathematics and acting (see below) — 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 Omaha, Nebraska who were asked to forward the letter to a stockbroker living at a specified location in Sharon, Massachusetts. The participants 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 50 people responded to the challenge, only three letters eventually reached their destination. Milgram's celebrated 1967 paper [1] refers to the fact that one of the letters in this initial experiment reached the recipient in just four days, but neglects to mention 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. The small world phenomenon is the theory that everyone in the world can be reached through a short chain of social acquaintances. ... Nickname: Gateway to the West Location in Nebraska Coordinates: Country United States State Nebraska County Douglas Founded 1854 Incorporated 1857 Mayor Michael Fahey Area    - City 307. ... Sharon (known to be the best town ever) is a town located in Norfolk County, Massachusetts. ... 1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar (the link is to a full 1967 calendar). ...

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 many 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 six 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." Michigan State University (MSU) is a public university in East Lansing, Michigan. ... This article describes some ethnic, historic, and cultural aspects of the Jewish identity; for a consideration of the Jewish religion, refer to the article Judaism. ... Motto: Concordia Salus Coordinates: Country Canada Province Quebec Founded 1642 Established 1832 City Mayor Gérald Tremblay Area    - City 366. ... This article or section is in need of attention from an expert on the subject. ... Six degrees of freedom (6DOF) refers to motion in three dimensional space, namely the ability to move forward/backward, up/down, left/right (translation in three perpendicular axes) combined with rotation about three perpendicular axes (yaw, pitch, roll). ...

One problem with conducting such a study, however, is that it assumes people in the chain are competent at discovering the connection between two people serving as end points.

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". 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. To meet Wikipedias quality standards, this article or section may require cleanup. ... This article or section does not cite its references or sources. ... The title given to this article is incorrect due to technical limitations. ... Paul Erdos Paul ErdÅ‘s (March 26, 1913 – September 20, 1996) was an immensely prolific and famously eccentric mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory. ... Autographed photo of Kevin Bacon greeting a crowd at Jefferson Square in Columbia, South Carolina in October, 2006 while filming Death Sentence. ... The trivia game Six Degrees of Kevin Bacon is based on a variation of the concept of the small world phenomenon which states that any actor can be linked, through their film roles, to Kevin Bacon. ... Go is a board game for two players. ... Honinbo Shusaku (本因坊秀策, Honinbō ShÅ«saku, born as Kuwabara Torajirō (桑原虎次郎 Kuwabara Torajirō), June 6, 1829 - August 10, 1862) was a professional Go player and is considered by many to be the greatest player of the golden age of Go in the mid-19th century. ... The Shusaku number represents the distance between a Go player and Honinbo Shusaku, measured in Go opponents. ...

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. The phrase tipping point or angle of repose is a sociology term that refers to that dramatic moment when something unique becomes common. ... Malcolm Gladwell Malcolm Gladwell (born September 1, 1963) is a United Kingdom-born, Canadian-raised journalist now based in New York City who has been a staff writer for The New Yorker since 1996. ... The New Yorker is an American magazine that publishes reportage, criticism, essays, cartoons, poetry and fiction. ... Connectors are people in a community who know large numbers of people and who are in the habit of making introductions. ...

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). I can't find this research Graph with SCC marked A directed graph is called strongly connected if for every pair of vertices u and v there is a path from u to v and a path from v to u. ...

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: 1998 (MCMXCVIII) was a common year starting on Thursday of the Gregorian calendar, and was designated the International Year of the Ocean. ... Duncan J. Watts is an associate professor of sociology at Columbia University and author of the book Six Degrees: The Science of a Connected Age (Norton, 2003). ... Steven H. Strogatz is Professor of theoretical and applied mechanics at Cornell University. ... Cornell redirects here. ... Binomial name Caenorhabditis elegans Wild-type C. elegans hermaphrodite stained to highlight the nuclei of all cells Caenorhabditis elegans () is a free-living nematode (a roundworm), about 1 mm in length, which lives in a temperate soil environment. ... Transmission towers Transmission lines in Lund, Sweden Electric power transmission, or more accurately Electrical energy transmission, is the second process in the delivery of electricity to consumers. ... Subfamilies See text Crickets, family Gryllidae (also known as true crickets), are insects somewhat related to grasshoppers and more closely related to katydids or bush crickets (order Orthoptera). ... Stridulation is the production of sounds by rubbing two parts of the body together; this mechanism is best known in crickets, grasshoppers, and cicadas. ...

"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." [2]

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. Mark Granovetter is a sociologist who gave some of the most influential theories in modern sociology, since the 1970s. ... A social network is a social structure made of nodes which are generally individuals or organizations. ... Jon Kleinberg is a Professor of Computer Science at Cornell University. ... In network science, which touches scientific disciplines as varied as computer science, mathematics, physics, biology and sociology, the term complex network refers to a network (graph) that has non-trivial topological structure. ... Network theory or diktyology is a branch of applied mathematics and physics, with the same general subject matter as graph theory. ... In mathematics and physics, a small-world network is a class of random graphs where most nodes are also neighbors of one another, but every node can be reached from every other by a small number of hops or steps. ... In mathematics, a random graph is a graph that is generated by some random process. ... A scale-free network is a specific kind of complex network in which the distribution of connectivity is extremely uneven. ...

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. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

See also

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

The Erdős number, honouring the late Hungarian mathematician Paul Erdős, one of the most prolific writers of mathematical papers, is a way of describing the collaborative distance, in regard to mathematical papers, between an author and Erdős. ... The Bacon number of an actor or actress is the number of degrees of separation (see Six degrees of separation) they have from actor Kevin Bacon, as defined by the game known as Six Degrees of Kevin Bacon. ... In mathematics, a random graph is a graph that is generated by some random process. ... In mathematics and physics, a small-world network is a class of random graphs where most nodes are also neighbors of one another, but every node can be reached from every other by a small number of hops or steps. ... A social network is a social structure made of nodes which are generally individuals or organizations. ... A scale-free network is a specific kind of complex network that has attracted attention since many real-world networks fall into this category. ...

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
• 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 Door - 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.