Why Your Friends Have More Friends Than You

Chances are you have fewer friends than most of your friends have. It might seem strange, but it's true for almost all of us. The phenomenon is called the "friendship paradox", and it's a mathematical pattern that underlies many of life's little annoyances.

Photo by Michael Sen Gallagher

Steven Strogatz at the New York Times explains that our perception is skewed by weighted averages, counting not just the number being averaged, but also the relative importance of each number. For example, for two classes, one with 90 students in it and the other with 10, the average class size should be 50, because (90 + 10)/2 = 50. However, most students (90 out of 100) would think the average class size is larger than 50 because many more of them experience the 90-student class.

This is the same phenomenon at work when you go to the gym and feel everyone else is more fit than you are (the gym skews towards gym rats) or when you think public places seem to be more crowded than the average would suggest.

When it comes to friendship and social networks, popular friends "contribute disproportionately to the average, since besides [having more friends], they're named as friends more frequently". Strogatz cites a recent study that examined 721 million Facebook users and found that a user's friend count was less than the average friend count of his/her friends 93 per cent of the time. The theory is that the average number of "friends of friends" will always be greater than the average number of friends of individuals, because the more popular people are counted more than once.

So don't worry if it seems like your friends are more popular than you are. We're all in the same boat.

Friends You Can Count On [New York Times]


    Wouldn't a weighted average for perception of class size make more sense?

    90 students have a class size of 90 = 8100 weighted size
    10 students ahve a class size of 10 = 100 weighted size

    Students class weighted class size is 82.

    Or just use mode of 90, rather then a decidely mean average.

    Hmmmm...it must be one of those days, because I'm just not getting it.

    So is it our perception that's at fault, or is there some mathematical quirk (that I don't understand) that somehow makes it possible for our friends to have more friends that us?

      I'm with you.

      From my understanding, it's all about the perception.

    This is certainly true for me. I'm not bothered by it in the least.

    From the NYT article, "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."
    I understand it this way: If I have 100 friends, I show up in 100 other people's friends lists.
    If my friend has 200 friends, they show up in 200 other friend lists.
    By this friend-friend list reciprocation, any individual with a larger number of friends, will automatically show up on a larger number of friend lists.
    The result is that people with more friends appear in more lists, this increased frequency of 'list inclusion' means that, on average, any one person's list is likely to have more people with high friend counts than the average friend count. Anyway, friend me, www.facebook.com/kendal

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