**What may look like a psychological phenomenon, is actually basic maths.**

In a colossal study of Facebook by Johan
Ugander, Brian Karrer, Lars Backstrom and Cameron Marlow, examined all of
Facebook’s active users, which at the time included 721 million people — about
10 percent of the world’s population — with 69 billion friendships among them. They
found that a user’s friend count was less than the average friend count of his
or her friends, 93 percent of the time. Next, they measured averages across
Facebook as a whole, and found that users had an average of 190 friends, while
their friends averaged 635 friends of their own.

Studies of offline social networks show
the same trend. It has nothing to do with personalities; it follows from basic
arithmetic. For any network where some people have more friends than others,
it’s a theorem that the average number of friends of friends is always
greater than the average number of friends of individuals.

This phenomenon has been called thefriendship paradox. Its explanation hinges on a numerical pattern — a
particular kind of “weighted average” — that comes up in many other situations.
Understanding that pattern will help you feel better about some of life’s
little annoyances.

In this hypothetical example, Ross, Chandler,
Phoebe and Rachel are four friends. Lines signify reciprocal friendships
between them; two people are connected if they’ve named each other as friends.

Ross’s only friend is Chandler, a social
butterfly who is friends with everyone. Phoebe and Rachel are friends with each
other and with Chandler. So Ross has 1 friend, Chandler has 3, Phoebe has 2 and
Rachel has 2. That adds up to 8 friends in total, and since there are 4 girls,
the average friend count is 2 friends per girl. This average, 2, represents the
“average number of friends of individuals” in the statement of the friendship
paradox. Remember, the paradox asserts that this number is smaller than the
“average number of friends of friends” — but is it? Part of what makes this
question so dizzying is its sing-song language. Repeatedly saying, writing, or
thinking about “friends of friends” can easily provoke nausea. So to avoid
that, I’ll define a friend’s “score” to be the number of friends she has. Then
the question becomes: What’s the average score of all the friends in the
network?

Imagine each person calling out the
scores of his/her friends. Meanwhile an accountant waits nearby to compute the
average of these scores.

Ross: “Chandler has a score of 3.”

Chandler: “Ross has a score of 1. Phoebe
has 2. Rachel has 2.”

Phoebe: “Chandler has 3. Rachel has 2.”

Rachel: “Chandler has 3. Phoebe has 2.”

These scores add up to 3 + 1 + 2 + 2 + 3
+ 2 + 3 + 2, which equals 18. Since 8 scores were called out, the average score
is 18 divided by 8, which equals 2.25.

Notice that 2.25 is greater than 2. The
friends on average

*do*have a higher score than the girls themselves. That’s what the friendship paradox said would happen.
The key point is

*why*this happens. It’s because popular friends like Chandler contribute disproportionately to the average, since besides having a high score, they’re also named as friends more frequently. Watch how this plays out in the sum that became 18 above: Ross was mentioned once, since she has a score of 1 (there was only 1 friend to call her name) and therefore she contributes a total of 1 x 1 to the sum; Chandler was mentioned 3 times because she has a score of 3, so she contributes 3 x 3; Phoebe and Rachel were each mentioned twice and contribute 2 each time, thus adding 2 x 2 apiece to the sum. Hence the total score of the friends is (1 x 1) + (3 x 3) + (2 x 2) + (2 x 2), and the corresponding average score is
Each individual’s score is multiplied by
itself before being summed. In other words, the scores are

*squared*before they’re added. That squaring operation gives extra weight to the largest numbers (like Chandler’s 3 in the example above) and thereby tilts the weighted average upward.
So that’s intuitively why friends have
more friends, on average, than individuals do. The friends’ average — a
weighted average boosted upward by the big squared terms — always beats the
individuals’ average, which isn’t weighted in this way.

Like many of math’s beautiful ideas, the
friendship paradox has led to exciting practical applications unforeseen by its
discoverers. It recently inspired an early-warning system for detecting
outbreaks of infectious diseases. In a study conducted at Harvard during
the H1N1 flu pandemic of 2009, the network scientists Nicholas Christakis and
James Fowler monitored the flu status of a large cohort of random undergraduates
and found that people with more connections were infected faster.

For more analogies check out the whole article at a New York Times blog.

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