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The Friendship Paradox: Your friends, on average, have more friends than you do

Dana Ernst
September 11, 2015

The Friendship Paradox: Your friends, on average, have more friends than you do

The Friendship Paradox is the observation that your friends, on average, have more friends than you do. This phenomenon, which was first observed by the sociologist Scott L. Feld in 1991, is mathematically provable. In this episode of FAMUS, we will discuss the "paradox", sketch its proof, and explore some applications. The idea for the talk was inspired by a post by Richard Green on Google+.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, September 11, 2015.

Dana Ernst

September 11, 2015
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  1. the friendship paradox: your friends, on
    average, have more friends than you do
    Friday Afternoon Mathematics Undergraduate Seminar
    Dana C. Ernst
    Northern Arizona University
    September 11, 2015

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  2. the friendship paradox
    The Paradox
    Despite your best efforts, you probably have fewer friends than
    most of your friends.
    But you aren’t special: the same is true for most of us.
    1

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  3. the friendship paradox
    Study of Facebook (2012)
    ∙ 721 million users (≈ 10% world’s population) ← vertices of graph
    ∙ 69 billion friendships ← edges of graph
    ∙ Researchers looked at a user’s friend count & compared to that
    user’s circle of friends
    ∙ 98% of the time: user’s friend count < average friend count of
    his/her friends
    ∙ Users had an average of 190 friends while their friends averaged
    635 friends
    It might seem counterintuitive that this is possible.
    2

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  4. the friendship paradox
    The findings of the Facebook study are not unusual & have been
    replicated in other social networks (e.g., Twitter) and in real life.
    For any network where some people have more “friends” than
    others, we have the following.
    Theorem (The Friendship Paradox, 1991)
    The average number of friends of individuals is always less than the
    average number of friends of friends.
    Explanation hinges on understanding a kind of “weighted average.”
    3

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  5. the friendship paradox
    Example 1
    Consider a professor that teaches two classes.
    ∙ Large class consisting of 90 students
    ∙ Small class consisting of 10 students
    What is average class size for this professor?
    90 + 10
    2
    = 50
    Not wrong, but misleading…
    4

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  6. the friendship paradox
    Example 1 (continued)
    Let’s consider the student point of view: 90/100 find themselves
    sitting in a big class while 10/100 experience a small class.
    Student-weighted average: How big is your class?
    ∙ 90 say 90
    ∙ 10 say 10
    Sum of responses:
    (90 × 90) + (10 × 10) = 8200
    Average class size that students experience:
    (90 × 90) + (10 × 10)
    90 + 10
    =
    8200
    100
    = 82 > 50
    5

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  7. the friendship paradox
    Example 2
    Consider the following friend network.
    B
    A
    C
    D
    What is average number of friends of individuals?
    1 + 3 + 2 + 2
    4
    =
    8
    4
    = 2
    The Friendship Paradox claims that this number is smaller than the
    average number of friends of friends.
    6

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  8. the friendship paradox
    Example 2 (continued)
    What is the average number of friends of friends? Ask each
    individual how many friends his/her friends have.
    A: B has a score of 3.
    B: A has a score of 1, C has a score of 2, D has a score of 2.
    C: B has a score of 3, D has a score of 2.
    D: B has a score of 3, C has a score of 2.
    Total number of friends of friends:
    3 + 1 + 2 + 2 + 3 + 2 + 3 + 2 = 18
    Average number of friends of friends:
    18
    8
    = 2.25 > 2
    7

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  9. the friendship paradox
    Example 2 (continued)
    Why does this happen? In addition to having a high score, popular
    friends contribute to other people’s scores more frequently. So,
    popular friends like B contribute disproportionately to the average.
    ∙ A mentioned 1 time since score of 1 → contributes 1 × 1
    ∙ B mentioned 3 times since score of 3 → contributes 3 × 3
    ∙ C mentioned 2 times since score of 2 → contributes 2 × 2
    ∙ D mentioned 2 times since score of 2 → contributes 2 × 2
    Weighted average of scores 1, 3, 2, 2 weighted by scores themselves:
    (1 × 1) + (3 × 3) + (2 × 2) + (2 × 2)
    1 + 3 + 2 + 2
    Big idea: Square terms are large!
    8

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  10. the friendship paradox
    Applications
    Strategy for detecting and combatting disease outbreaks.
    ∙ Monitor/immunize friends of random individuals instead of
    random individuals.
    ∙ One study got two-week lead time on H1N1.
    ∙ Model: If random individuals are immunized, 80–90% of
    population must be immunized to attain herd immunity. However,
    herd immunity achieved when 20–40% of friend population
    immunized.
    Your girl(boy)friend has likely kissed more people than you.
    9

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  11. the friendship paradox
    Majority Illusion
    The recent paper The Majority Illusion in Social Networks (2015)
    explains how, under certain conditions, the structure of a social
    network can make it appear to an individual that certain types of
    behavior are far more common than they actually are.
    10

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  12. the friendship paradox
    Majority Illusion (continued)
    Consider the following networks of friends, (a) and (b), each
    containing 14 individuals, where red corresponds to heavy drinker.
    (a) Heavy drinkers are 3 of the most popular people; each of the
    other 11 individuals observes that at least half of their friends are
    heavy drinkers, which leads them to think that heavy drinking is
    common. But: Only 3/14 of the group are heavy drinkers.
    (b) Heavy drinkers are not particularly popular, and nobody in group
    has heavy drinkers as most of their friends.
    11

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  13. the friendship paradox
    References
    ∙ Richard Green’s post on Google+ about the Friendship Paradox
    inspired me to learn more about this topic.
    ∙ A lot of the content of my slides came directly from the article
    Friends You Can Count On by Stephen Strogatz at NY Times
    Opinator.
    ∙ The Wikipedia article on the Friendship Paradox is excellent and
    contains a sketch of the proof of the paradox.
    ∙ The article The Majority Illusion in Social Networks (2015) by
    Lerman, Yan, Wu explores some phenomena that are related to the
    Friendship Paradox.
    12

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