Small world experiment
Encyclopedia
The small world experiment comprised several experiments conducted by Stanley Milgram
and other researchers examining the average path length
for social network
s 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.
's conjectures based on his radio work in the early 20th century, which were articulated in his 1909 Nobel Prize
address, may have inspired Hungarian author Frigyes Karinthy
to write a challenge to find another person to whom he could not be connected through at most five people. 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 Their unpublished manuscript circulated among academics for over 20 years before publication in 1978.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 network
s, 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 network
s.
Milgram took up the challenge on his return from Paris, leading to the experiments reported in "The Small World Problem" in 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 Austria
n 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 slower 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 works is a study of obedience and authority, which is widely known as the Milgram Experiment
. 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 network
s.
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.
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.
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.
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.
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. This suggests that participants strongly favored geographic characteristics when choosing an appropriate next person in the chain.
In addition to these methodological critiques, 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).
by Malcolm Gladwell
, based on articles originally published in The New Yorker
, elaborates 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).
s and actor
s, 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.
. 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.
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 the neural network of C. elegans
and power grids, 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:
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 network
s. 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.
has become part of the collective consciousness. Social networking websites such as Facebook
, Friendster
, MySpace
, XING
, Orkut
, Cyworld
, Bebo
, and others have greatly increased the connectivity of the online space through the application of social network
ing concepts. The potential of the small world effect in linking likely but unknown collaborators using social networking was pointed out explicitly in The IRG Solution – hierarchical incompetence and how to overcome it in 1984.
Stanley Milgram
Stanley Milgram was an American social psychologist most notable for his controversial study known as the Milgram Experiment. The study was conducted in the 1960s during Milgram's professorship at Yale...
and other researchers examining the average path length
Average path length
Average path length is a concept in network topology that is defined as the average number of steps along the shortest paths for all possible pairs of network nodes...
for social network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
s of people in the United States. The research was groundbreaking in that it suggested that human society is a small world
Watts and Strogatz model
The Watts and Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their joint 1998 Nature paper...
type network characterized by short path lengths. The experiments are often associated with the phrase "six degrees of separation
Six degrees of separation
Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer...
", although Milgram did not use this term himself.
Historical context of the small world problem
Guglielmo MarconiGuglielmo Marconi
Guglielmo Marconi was an Italian inventor, known as the father of long distance radio transmission and for his development of Marconi's law and a radio telegraph system. Marconi is often credited as the inventor of radio, and indeed he shared the 1909 Nobel Prize in Physics with Karl Ferdinand...
's conjectures based on his radio work in the early 20th century, which were articulated in his 1909 Nobel Prize
Nobel Prize
The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895...
address, may have inspired Hungarian author Frigyes Karinthy
Frigyes Karinthy
Frigyes Karinthy was a Hungarian author, playwright, poet, journalist, and translator. He was the first proponent of the six degrees of separation concept, in his 1929 short story, Chains . Karinthy remains one of the most popular Hungarian writers...
to write a challenge to find another person to whom he could not be connected through at most five people. This is perhaps the earliest reference to the concept of six degrees of separation
Six degrees of separation
Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer...
, and the search for an answer to the small world problem.
Mathematician Manfred Kochen and political scientist Ithiel de Sola Pool
Ithiel de Sola Pool
Ithiel de Sola Pool was a revolutionary in the field of social sciences. Pool led groundbreaking research on technology and its effects on society. He coined the term "convergence" to describe the effect of various scientific innovations on society in a futuristic world...
wrote a mathematical manuscript, "Contacts and Influences", while working at the University of Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
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 network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
s, 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 network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
s.
Milgram took up the challenge on his return from Paris, leading to the experiments reported in "The Small World Problem" in May 1967 (charter) issue of the popular magazine Psychology Today
Psychology Today
Psychology Today is a bi-monthly magazine published in the United States. It is a psychology-based magazine about relationships, health, and related topics written for a mass audience of non-psychologists. Psychology Today was founded in 1967 and features articles on such topics as love,...
, with a more rigorous version of the paper appearing in Sociometry
Sociometry
Sociometry is a quantitative method for measuring social relationships. It was developed by psychotherapist Jacob L. Moreno in his studies of the relationship between social structures and psychological well-being....
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 Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n who had been involved in Statist urban design
Urban design
Urban design concerns the arrangement, appearance and functionality of towns and cities, and in particular the shaping and uses of urban public space. It has traditionally been regarded as a disciplinary subset of urban planning, landscape architecture, or architecture and in more recent times has...
, 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 slower 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
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
. One of Milgram's most famous works is a study of obedience and authority, which is widely known as the Milgram Experiment
Milgram experiment
The Milgram experiment on obedience to authority figures was a series of notable social psychology experiments conducted by Yale University psychologist Stanley Milgram, which measured the willingness of study participants to obey an authority figure who instructed them to perform acts that...
. 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
Frigyes Karinthy
Frigyes Karinthy was a Hungarian author, playwright, poet, journalist, and translator. He was the first proponent of the six degrees of separation concept, in his 1929 short story, Chains . Karinthy remains one of the most popular Hungarian writers...
in the 1920s while documenting a widely circulated belief in Budapest
Budapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
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
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
, born in Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
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
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
and later IBM
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
in the U.S.). This circle of researchers was fascinated by the interconnectedness and "social capital" of social network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
s.
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
Psychology Today
Psychology Today is a bi-monthly magazine published in the United States. It is a psychology-based magazine about relationships, health, and related topics written for a mass audience of non-psychologists. Psychology Today was founded in 1967 and features articles on such topics as love,...
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
John Guare
John Guare is an American playwright. He is best known as the author of The House of Blue Leaves, Six Degrees of Separation, and Landscape of the Body...
, 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. 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 networkSocial network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
and attempt to find the average path length
Average path length
Average path length is a concept in network topology that is defined as the average number of steps along the shortest paths for all possible pairs of network nodes...
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
- Though the experiment went through several variations, Milgram typically chose individuals in the U.S. cities of Omaha, NebraskaOmaha, NebraskaOmaha is the largest city in the state of Nebraska, United States, and is the county seat of Douglas County. It is located in the Midwestern United States on the Missouri River, about 20 miles north of the mouth of the Platte River...
and Wichita, KansasWichita, KansasWichita is the largest city in the U.S. state of Kansas.As of the 2010 census, the city population was 382,368. Located in south-central Kansas on the Arkansas River, Wichita is the county seat of Sedgwick County and the principal city of the Wichita metropolitan area...
to be the starting points and Boston, Massachusetts to be the end point of a chainChain letterA typical chain letter consists of a message that attempts to the recipient to make a number of copies of the letter and then pass them on to as many recipients as possible...
of correspondence. These cities were selected because they were thought to represent a great distance in the United States, both socially and geographically. - 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.
- 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.
- 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 he knew personally who was more likely to know the target. He was then directed to sign his 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.
- 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.However, 64 of the letters eventually did reach the target contact. Among these chains, the average path length
Average path length
Average path length is a concept in network topology that is defined as the average number of steps along the shortest paths for all possible pairs of network nodes...
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
Six degrees of separation
Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer...
", these findings are likely to have contributed to its widespread acceptance.
In an experiment in which 160 letters were mailed out, 24 reached the target in his Sharon, Massachusetts
Sharon, Massachusetts
Sharon is a town in Norfolk County, Massachusetts, United States. The population was 17,612 at the 2010 census. Sharon is part of Greater Boston, about 17 miles southwest of downtown Boston....
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.
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. 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:- 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. First, the "starters" were not chosen at random, as they were recruited through an advertisement that specifically sought for people who considered themselves as 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 under-estimate the true average path length. Several methods have been suggested to correct these estimates; one uses a variant of survival analysisSurvival analysisSurvival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
in order to account for the length information of interrupted chains, and thus reduce the bias in the estimation of average degrees of separation. - 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 bias and over-estimate the average number of ties needed for two random people.
- 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 StrogatzSteven StrogatzSteven Henry Strogatz is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University...
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 AgeSix Degrees: The Science of a Connected AgeSix Degrees: the science of a connected age is a popular science book by Duncan J...
. - 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 critiques, 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).
The social sciences
The Tipping PointThe Tipping Point
The Tipping Point: How Little Things Can Make a Big Difference is a book by Malcolm Gladwell, first published by Little Brown in 2000....
by Malcolm Gladwell
Malcolm Gladwell
Malcolm Gladwell, CM is a Canadian journalist, bestselling author, and speaker. He is currently based in New York City and has been a staff writer for The New Yorker since 1996...
, based on articles originally published in The New Yorker
The New Yorker
The New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons and poetry published by Condé Nast...
, elaborates 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
Strongly connected component
A directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex. In particular, this means paths in each direction; a path from a to b and also a path from b to a....
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
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
(Barrett et al., 2005).
Mathematicians and actors
Smaller communities, such as mathematicianMathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s and actor
Actor
An actor is a person who acts in a dramatic production and who works in film, television, theatre, or radio in that capacity...
s, have been found to be densely connected by chains of personal or professional associations. Mathematicians have created the Erdős number
Erdos number
The Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.The same principle has been proposed for other eminent persons in other fields.- Overview :...
to describe their distance from Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
based on shared publications. A similar exercise has been carried out for the actor Kevin Bacon
Kevin Bacon
Kevin Norwood Bacon is an American film and theater actor whose notable roles include Animal House, Diner, Footloose, Flatliners, Wild Things, A Few Good Men, JFK, Apollo 13, Mystic River, The Woodsman, Trapped, Friday the 13th, Hollow Man, Tremors, Death Sentence, Frost/Nixon, Crazy, Stupid, Love....
and other actors who appeared in movies together with him — the latter effort informing the game "Six Degrees of Kevin Bacon
Six Degrees of Kevin Bacon
Six Degrees of Kevin Bacon is a trivia game based on the concept of the small world phenomenon and rests on the assumption that any individual involved in the Hollywood, California film industry can be linked through his or her film roles to actor Kevin Bacon within six steps. The name of the game...
". There is also the combined Erdős-Bacon number
Erdos-Bacon number
A person's Erdős–Bacon number is the sum of one's Erdős number—which measures the "collaborative distance" in authoring mathematical papers between that person and Hungarian mathematician Paul Erdős—and one's Bacon number—which represents the number of links, through roles in...
, for actor-mathematicians and mathematician-actors. Players of the popular Asian game Go
Go (board game)
Go , is an ancient board game for two players that originated in China more than 2,000 years ago...
describe their distance from the great player Honinbo Shusaku
Honinbo Shusaku
Honinbo Shusaku 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.- Biography :He was nicknamed "Invincible" after he earned a perfect score for 19 straight wins in the annual castle...
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, Peter Dodds, Roby Muhamad, and Duncan Watts conducted the first large-scale replication of Milgram's experiment, involving 24,163 e-mail chains and 18 targets around the world.http://www.sciencemag.org/cgi/content/full/301/5634/827 Dodds et al. also found that the mean chain length was roughly six, even after accounting for attrition. A similar experiment using popular social networking sites as a medium was carried out at Carnegie Mellon UniversityCarnegie Mellon University
Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania, United States....
. 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. WattsDuncan J. Watts
Duncan J. Watts is an Australian researcher and a principal research scientist at Yahoo! Research, where he directs the Human Social Dynamics group. He is also a past external faculty member of the Santa Fe Institute and a former professor of sociology at Columbia University, where he headed the...
and Steven Strogatz
Steven Strogatz
Steven Henry Strogatz is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University...
from Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
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
Caenorhabditis elegans
Caenorhabditis elegans is a free-living, transparent nematode , about 1 mm in length, which lives in temperate soil environments. Research into the molecular and developmental biology of C. elegans was begun in 1974 by Sydney Brenner and it has since been used extensively as a model...
and power grids, 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
Cricket (insect)
Crickets, family Gryllidae , are insects somewhat related to grasshoppers, and more closely related to katydids or bush crickets . They have somewhat flattened bodies and long antennae. There are about 900 species of crickets...
chirps
Stridulation
Stridulation is the act of producing sound by rubbing together certain body parts. This behavior is mostly associated with insects, but other animals are known to do this as well, such as a number of species of fishes, snakes and spiders...
, 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."
Generally, their model demonstrated the truth in Mark Granovetter
Mark Granovetter
Professor Mark Granovetter is an American sociologist at Stanford University who has created theories in modern sociology since the 1970s. He is best known for his work in social network theory and in economic sociology, particularly his theory on the spread of information in social networks known...
's observation that it is "the strength of weak ties" that holds together a social network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
. Although the specific model has since been generalized by Jon Kleinberg
Jon Kleinberg
-External links:**** Stephen Ibaraki*Yury Lifshits,...
, it remains a canonical case study in the field of complex network
Complex network
In the context of network theory, a complex network is a graph with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in real graphs...
s. In network theory
Network theory
Network theory is an area of computer science and network science and part of graph theory. It has application in many disciplines including statistical physics, particle physics, computer science, biology, economics, operations research, and sociology...
, the idea presented in the small-world network
Small-world network
In mathematics, physics and sociology, a small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps...
model has been explored quite extensively. Indeed, several classic results in random graph
Random graph
In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...
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
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, 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 degreesSix degrees of separation
Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer...
has become part of the collective consciousness. Social networking websites such as Facebook
Facebook
Facebook is a social networking service and website launched in February 2004, operated and privately owned by Facebook, Inc. , Facebook has more than 800 million active users. Users must register before using the site, after which they may create a personal profile, add other users as...
, Friendster
Friendster
Friendster is a social gaming site that is based in Malaysia, KL. The company now operates mainly from the three Asian countries namely in the Philippines, Malaysia and Singapore....
, MySpace
MySpace
Myspace is a social networking service owned by Specific Media LLC and pop star Justin Timberlake. Myspace launched in August 2003 and is headquartered in Beverly Hills, California. In August 2011, Myspace had 33.1 million unique U.S. visitors....
Xing
Xing may refer to:* an abbreviation for crossing, primarily used in North America* Qiao Xing Universal Telephone Inc. * XING, a social network platform* Xing County, in Shanxi, China* Xing - A Korean boyband...
, Orkut
Orkut
Orkut is a social networking website that is owned and operated by Google Inc. The service is designed to help users meet new and old friends and maintain existing relationships...
, Cyworld
Cyworld
Cyworld is a South Korean social network service operated by SK Communications , a subsidiary of SK Telecom .Members cultivate relationships by forming Ilchon or "friendships" with each other through their minihompy....
, Bebo
Bebo
Bebo is a social networking website launched in July 2005. It is currently owned and operated by Criterion Capital Partners after taking over from AOL in June 2010....
, and others have greatly increased the connectivity of the online space through the application of social network
Social network
A social network is a social structure made up of individuals called "nodes", which are tied by one or more specific types of interdependency, such as friendship, kinship, common interest, financial exchange, dislike, sexual relationships, or relationships of beliefs, knowledge or prestige.Social...
ing concepts. The potential of the small world effect in linking likely but unknown collaborators using social networking was pointed out explicitly in The IRG Solution – hierarchical incompetence and how to overcome it in 1984.
See also
- Bacon number
- Erdős numberErdos numberThe Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.The same principle has been proposed for other eminent persons in other fields.- Overview :...
- Erdős–Bacon number
- Personal NetworkPersonal NetworkA Personal Network is a set of human contacts known to an individual, with whom that individual would expect to interact at intervals to support a given set of activities....
- Random network
- Richard GilliamRichard GilliamRichard Gilliam is a short story author and the editor of such theme anthologies as Confederacy of the Dead , Phobias and the Grails series...
External links
- The Small World Experiment - 54 little boxes travelling the world
- What the Milgram Papers in the Yale Archives Reveal About the Original Small World Study (broken link)
- Planetary-Scale Views on an Instant-Messaging Network
- 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
- - 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.