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The Friendship Paradox

June 17, 2014

It seems to be in our nature to compare our accomplishments to those of others. Teenagers worry about popularity. Later in life we compare our success to that of friends. But, there is a mathematical reason why we usually come up short!

You may have noticed that your friends seem to have more friends than you do.  And you are right – on average, your friends are more popular than you are. This is true on Facebook and in real life, and is a consequence of what statisticians call biased sampling – you are more likely to befriend an outgoing, easy-to-get-along person than a recluse who hardly talks to anyone. Your typical friends make friends easily. Therefore you do not form friendships at random – statisticians would say that you are taking a biased sample of society.

Biased sampling is why playing a game of poker against strangers in a casino is usually a bad idea – you are more likely to meet an opponent who spends a lot of time playing the game, rather than a beginner or a complete amateur.

More surprisingly, not only are your friends more popular, but on average they are also more successful. The reason for this is that people with more friends seem to be on average more successful. We already established that because of biased sampling your friends have more friends than you do. If more friends means more success, it follows that your friends are on average more successful than you are. Hence comparing yourself to your friends, is not a good idea – you are using a biased sample of society that is likely doing better than you.

OK, you may say, but is it useful?  Indeed it can be – select a group of students at a university, and ask them to give you the name of a few of their friends.  On average they will name people who are more popular, and therefore have more social contacts than the average student. If you are interested in hearing the latest rumor, you will be well advised to go to this new group. But, since these friends are more popular, they also have more interactions with others, and may be among the first to get sick in an epidemic.

Researchers have confirmed that this is the case: they asked random students to name friends, and found that in a flu outbreak this named group got sick about two weeks earlier than the average student. To get an early warning of an epidemic, just pick people at random and ask them if their friends are sick. You can do the same if you are trying to spot a new trend.

So mathematics tells us something valuable about our friends:  Comparing our accomplishments to theirs is likely going to leave us depressed. Instead listen to your friends if you want to hear about a good place to eat, what concert to go to, or interesting new technology.  Your friends will be able to tell you about it better than the average person.

Some references: 

Popular articles about the friendship paradox can be found here.  Here is a more detailed, yet very easy to follow discussion by the inimitable Steve Strogatz.

The paradox was originally described by Scott L. Feld in  “Why your friends have more friends than you do,” American Journal of Sociology, Vol. 96:6, pp. 1464–1477 (1991).

The generalized friendship paradox – the observations that your friends are more successful than you are (on average), is described here. I have taken some liberties with the term success – the fact that friends are more successful has been shown, for example, for the number of co-authors and citations for scientific papers and and followers on Twitter. If this counts as “success” – and in scientific circles, then what I have said is strictly true.  However, it is also likely that the observation extends.

The article that describes how friends of friends can be used to track outbreaks of diseases is Christakis NA, Fowler JH (2010) Social Network Sensors for Early Detection of Contagious Outbreaks. PLoS ONE 5(9): e12948. doi:10.1371/ journal.pone.0012948, and can be found at    and a followup Garcia-Herranz M, Moro E, Cebrian M, Christakis NA, Fowler JH (2014) Using Friends as Sensors to Detect Global-Scale Contagious Outbreaks. PLoS ONE 9(4): e92413.

There is a difference between the mean and the median of the number of friends that I did not get into.  This is described in more detail here. This also provides a more detailed description of the mechanism behind the generalized friendship paradox.

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