Six Degrees of Separation - Fact or Fiction?

Would you believe it if I said that you could discover a connection to just about any unrelated person on the face of the Earth? Or that your favorite movie star or a notorious gangster can be connected to you through a chain of merely six people. Stunned? Brush through this article to find out how...
Anuja Marathe Kanhere
Six degrees of separation refers to a theory which states that each person on the face of the Earth is connected to all the other people through a chain or connection of not more than six people. This theory not only strives to explore connections between two remotely located people, but also aims to understand levels of social interaction, connectivity of people using the W3 or World Wide Web, and spread of certain epidemic diseases on account of social connections. Till date, there has been a lot of research on this subject to verify preciseness of this theory. Researchers feel that popular social networking sites like Facebook may have some answers, on account of their huge membership. Let us take a look at this unique theory, which has generated a lot of interest and hype among people ever since it was proposed.
Origin of The Theory
The six degrees theory was first proposed by Frigyes Karinthy, a Hungarian writer, in the year 1929. In his book titled 'Everything is Different', he first mentioned an abstract link or chain between people all across the world. To stress on this concept, he wrote in a game that his book characters played in the story. Here is a short extract of that story for your reference.
"A fascinating game grew out of this discussion. One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth - anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances."
The concept states that each person can be a friend of a friend or a relative of a relative. This can be explained with the diagram given below:
A----B----C----D----E----F----G

0----1----2----3----4----5----6

• A person named A wants to find a connection with a person named G. To simplify the matter, A is placed in position '0' and G in position '6'. There is supposed to be a link of 5 people in between A and G.
• All the people in this chain become related to the first and last person because of their connection to each other, which means B, C, D, E and F are related to A and G directly or indirectly.
• Individuals in the chain have an immediate connection with only those people who are on either sides in the chain. In the above diagram, A has an immediate connection with only B, while G has an immediate connection with only F. However, B is connected to A and C. The same applies to other individuals in the chain.
• The chain of links should pass from A to G.
Small World Experiment
Although the original theory suggested the number of links in the chain to be six, several experiments in this field have proved otherwise. One such experiment was popularly carried out by an American psychologist and Harvard University professor Stanley Milgram in the year 1967. Milgram's aim was to study the average number of links between two unrelated Americans using the principles of the Six Degrees of Separation Theory.
• As part of the experiment, Milgram randomly selected two individuals, a woman and a stock broker from Sharon, a suburb in the city of Boston, Massachusetts. These two individuals were the targets or the destination points for the chain links.
• The starting points of the experiment were randomly selected people residing in Omaha, Nebraska and Wichita, Kansas.
• Milgram distributed approximately 296 packets bound for one target and about 160 letters bound for the other. These packets and letters contained details of the theory, the nature of Milgram's experiment and details about the target persons. Additionally, they contained a log sheet and pre-addressed postcards to be sent to Milgram in Harvard by each recipient of the packets.
• The residents of Omaha and Wichita, who received the packets, were first asked whether they directly knew the target residing in Boston. If not, they were then asked to forward the packet or letter to any person of their choice, who according to them might know the target.
• Before forwarding the packets, recipients were asked to make a note of their own names on the log sheet and put it back inside the packet. They also needed to send one pre-addressed postcard with their own details to Milgram.
• This cycle of forwarding the packet was supposed to go on from one recipient to the next, until the packets reached the hands of the targets.
Conclusions
• Milgram observed that several participants of this experiment showed a reluctance to forward their letters or parcels to the next person. Out of the 296 parcels bound for Boston, only 64 reached the target. Similarly, out of the 160 letters that were originally mailed, merely 24 letters made it to their destination.
• Milgram's observations are therefore, based only on a small fraction of the total parcels originally distributed.
• The participants in this experiment gave tremendous importance to geographical proximity of the target persons. Which means, the parcels reached Boston very fast but moved at a slow pace within the city until they reached the targets.
• The log sheet and postcards received by Milgram showed that the number of links in a chain were anything ranging from nine to a meager two or three.
• As per Milgram, even though the original six degrees theory states a total of six links, it actually refers to maximum number of links expected to form a chain. One may come across a chain that has lesser than six links.
• Milgram concluded that the average degrees of separation between two unknown people staying the US was just 3 to 3.5 and not 6 as stated in the original theory. This is because Milgram narrowed his research within boundaries of United States and did not mention the degrees of separation between two people residing in other nations.
Small World Project
(This should not be confused with Stanley Milgram's Small World Experiment.)

Measuring degrees of separation between people living worldwide is a massive and difficult exercise. In spite of this, Prof. Duncan Watts from Columbia University made an attempt to find the average number of links using emails.
• In the year 2003 and 2004, Watts randomly selected 18 target people from about 13 nations around the globe.
• The starting points in Columbia's Small World Project were volunteers that were ready to send emails to their respective friends.
• Participants were asked to send a pre-drafted email to all people known to them.
• Subsequent recipients were also asked to forward the email to every person in their contact list. Thus, several email chains for the project started running on parallel basis.
• A particular email chain would stop as soon as one of 18 target recipients received the email.
Conclusions
• According to Watts, the project received huge response with approximately 60,000 email users from about 170 nations around the world taking part in the project.
• Although a chain of 24,000 emails got created during the project, just about 400 emails reached the targets. As per Watts, this might have happened due to the unwillingness of subsequent recipients to forward the emails.
• The successfully completed chains comprised an average of four links by the time they reached their targets.
• Watts factored the number of participants who quit the email chains and calculated the median length of an email chain to be approximately 5 and 7 persons.
• In the end, Watts did support the original theory by stating the average number of links to be 6.
Prof. Duncan Watts and Prof. Steven Strogatz from Cornell University came up with a mathematical formula to calculate the average degrees of separation between two people. The formula states,

▶ DS = log N/ log K

Where, DS refers to the degrees of separation
N refers to the total number of people capable of creating direct or indirect contact with others
K refers to the average number of friends or relations of each person.
Note: Prof. Watts used logarithms to simplify calculations for the massive data.

Example: Assume N comprises 90% of American population, which equals to 300,000,000, (the other 10% population comprise young children, who are yet to develop friends or contact links), and assume K has an average of 30 relations. Using the formula, we get (after applying logarithms)

DS = log (3*108) / log (30)
DS = 8.4771 / 1.4771
DS = 5.739
Therefore, DS = 6 (approx.)
It's not 6 Degrees, it's 4.74!
It is the weak ties that make the world small. ~Dr Jon Kleinberg, Cornell University~

There are plenty of social networking sites that have gained world-wide popularity today. Sites such as Facebook, Twitter, LinkedIn etc., boast of millions of worldwide user profiles. Avid users of Facebook will agree that this website springs some surprises when members find mutual friends on the most unexpected profiles. In fact, to popularize this theory, Facebook even launched a web application named Six Degrees. It prompts users to click on the 'Search for Someone' tab and type a name of any other Facebook user. This application can help users to trace their connection to that person and generate a report stating the number of links involved.

The Facebook team, in collaboration with the University of Milan, conducted a research on Degrees of Separation between its worldwide users. Here is a highlight on its crucial findings.
• Facebook analyzed its membership and the size of friendships between its members. Reports show that Facebook's 721 million active users have approximately 69 billion friendship links between them. This makes Facebook the largest relationship/ friendship related database in the whole world.
• Facebook's analysis team feels that people of a certain age tend to have friends of the same age range. Similarly, members of the website generally have 84% friends residing in the same country as theirs.
• As per Facebook, approximately 10% of their members have lesser than 10 friends. Another 20% have up to 25 friends only. However, 50% of Facebook users have more than 100 people on their friends list.
• The degrees of separation between two people residing in far off countries seems to reduce if the chain of links passes through multicultural and cosmopolitan countries like the US or the UK.
Conclusions
• According to Facebook's research, 92% of its members have only 4 degrees of separation.
• About 99.6% of other people have 5 degrees of separation.
• The degrees of separation seem to shrink if you are searching for a link between two people residing in the same country. At such times, just 3 degrees of separation might exist.
• The average degrees of separation between Facebook's worldwide users is just 4.74. This number is expected to reduce in the coming years, due to a consistent rise in Facebook users. This is because connections between people broaden with new friendships.
Many other social networking sites report the links between their users to range anywhere from 3 to 5.
Some Interesting Asides
The Oracle of Bacon
The Six Degrees concept has been playfully used to generate the Bacon number under the Oracle of Bacon. This number can be generated by deriving the degrees of separation a movie star has from actor Kevin Bacon. Bacon himself is always placed at number '0'. Actors who have directly worked with him are at number 1. People who have worked with actors who have worked with Bacon, are placed at number 2. This is how the chain continues. On account of Bacon's links to almost 90% of Hollywood stars, he is often quoted to be the Center of Hollywood Universe.

The Erdős Number
This peculiar idea of generating an Erdős number was first expressed by an analyst named Casper Goffman and popularized by renowned American mathematician, Ronald Graham. This number expresses the degrees of collaborative separation a researcher has from the world-famous mathematician Paul Erdős. It is said that Erdős wrote more than 1500 research papers during his lifetime. Out of these, some 511 papers were written in direct collaboration with his students and colleagues. As per the concept, Prof. Erdős was placed at number '0' since the collaborations were to be traced to him. His immediate and direct collaborators were placed at number '1'. People who collaborated with direct collaborators of Erdős, were allotted number '2'. As of 2011, people with Erdős number '2' were estimated to be 9267! The Erdős number is expected to rise indefinitely as the indirect collaborations increase each passing year.

In honor of some of the greatest legends in the fields of Science, Arts and Sports, people have come up with a few more unique numbers. These include the Einstein number (Albert Einstein, Physicist), Chomsky number (Noam Chomsky, Linguist), Morphy number (Paul Morphy, Chess master), and Vanderbilt number (Harold Vanderbilt, legendary Bridge player), in addition to others.

The world is smaller than you ever thought it to be and it is shrinking rapidly even as you blink! Although it is true that a connection can be traced between any two people on the globe, it is still not proven that the exact connection can be made by just six people. Most research results suggest average links to be anywhere between 2 and 9. So, until we perfect our calculations, we can safely conclude that contents of this theory are partly factual.