### Video Transcript

Six Degrees of Separation: Is It a
Small World After All?

This is Maisie. Maisie is a Springer Spaniel. Maisie is owned by a little girl
named Grace. Grace’s best school friend is
called AJ. And AJ’s cousin is a barista. One of AJ’s most regular customers
is a man named Raj. Raj happens to be married to
Carla. And well, Carla provides PR to a
well-known talent show mogul. There are seven individuals in this
chain. One, two, three, four, five who sit
between Maisie and the talent show mogul. Let’s call him Simon. But there are actually six links,
or connections, in the chain. And so, we say that there are six
degrees of separation between Maisie and Simon.

Now, the idea that anyone on the
planet is connected to any other person in this way is not a recent one. In 1929, Hungarian writer Frigyes
Karinthy postulated in his short story chains that the modern world was shrinking
due to the ever-increasing connectedness of human beings. The first person to study this
theory in any sort of scientific way was social psychologist Stanley Milgram.

A well-known and somewhat
controversial figure, Milgram carried out a series of experiments in which he
measured how likely people were to obey an authority figure by inflicting pain on
another person. He found that the more degrees of
separation there were, the more likely it was that the person would be willing to
inflict harm and even death on another.

Then, in 1969, he partnered with
Jeffrey Travers to carry out an experiment whereby they challenged 296 people in
Nebraska and Boston to send a letter through acquaintances, someone they knew on a
first-name basis only, to a target person in Massachusetts. Of the 64 letters that reached this
person, the mean number of links were 5.2, meaning that the average number of
degrees of separation was 6.2. Naturally, the validation of the
idea of six degrees of separation from this experiment led to a load of work aimed
at answering the question “why should there exist short chains of acquaintances
linking together arbitrary pairs of strangers?”

Well, a decade later, one
mathematician named Paul Erdos looked to mathematically simulate the idea using a
network of nodes which were randomly connected by links. He found that when the number of
links per node was small, the network is fragmented; there are very few common
connections. But if you exceed an average of one
connection per node, they almost all link up, forming a giant cluster of nodes with
lots of common connections.

Watts and Strogatz formalized this
and found that the average path length in a random network is equal to the natural
log of 𝑁 over the natural log of 𝑘, where 𝑁 is the total number of nodes and 𝑘
is the number of acquaintances per node.

For instance, if we take 90 percent
of the population of the United Kingdom to be approximately 60 million people — this
is, of course, assuming that 10 percent are too young to participate — and the
average number of acquaintances per person to be 40. So, 𝑁 is 60 million, and 𝑘 is
40. The degrees of separation is equal
to the natural log of six times 10 to the seventh power over the natural log of 40,
which is 4.86, correct to three significant figures.

If we complete this for the
population of the United States, where 𝑁 is approximately 300 million and 𝑘 is
equal to 40. We obtain the degrees of separation
to be the natural log of three times 10 to the eighth power divided by the natural
log of 40, which is 5.29. And it was also shown that the path
length grows extremely slowly as the network size increases.

So, taking 90 percent of the
population of the world to be six billion people and, once again, 𝑘, the number of
acquaintances per person, to be 40, we obtain the degrees of separation here to be
equal to 6.1. This is called a small-world
network. It’s a type of mathematical graph
in which most nodes are not neighbors of one another but in which most nodes can be
reached from every other by a small number of hops or steps.

For example, we see that this node
is not a neighbor to this node, but they can reach each other by completing one,
two, three steps. Outside of mathematics, this
small-world idea is heavily referenced in pop culture, from films and music through
to a full website which helps you establish the Bacon number of any Hollywood
celebrity. That’s the number of degrees of
separation that this celebrity has from Kevin Bacon himself.

But what effect has the invention
of social media had on this phenomenon? Let’s imagine you have 100 friends
on your social media page. Then, each of your friends has 100
friends. And each of their friends has 100
friends, and so on. Using the average-path-length
formula that Watts and Strogatz proposed. We find the number of degrees of
separation between you and every other person on Earth to be the natural log of six
times 10 to the ninth power divided by the natural log of 100, which is 4.89.

Increase that to 200 friends, and
this drops to 4.25. In fact, the average number of
social media friends is 338, at which point the degrees of separation drop further
to just 3.87. This phenomenon explains how videos
are able to go viral on social media. Remember when a video of a yodeling
boy in Walmart went viral after just one tweet? The video was retweeted 50,000
times in the space of less than two weeks launching Mason Ramsey into an album
deal. And so, it does appear that not
only, “Is it a very small world after all?”, but that it’s actually shrinking.