http://opinionator.blogs.nytimes.com/2012/09/17/friends-you-can-count-on/

The friendship paradox is a statistical phenomenon where the number of friends a person has is probably less than the average number of friends that their friends have. To show an example of this, we can analyze a simple example where there are three people A, B, and C. B is friends with both A and C, but A and C are not friends. If we take the average number of friends for each person, we get 4/3. However, if we take the average number of friends of friends, we get 5/3.

A recent study examined all of Facebook’s active users and found that Facebook users have an average of 190 friends, while their friends average 635 friends of their own. At first, this seems very odd. Some may think that this is due to sociology, but it is actually an arithmetic theorem that for any network where some people have more friends than others, the average number of friends of friends is always greater than the average number of friends of individuals.

This is because the friends of friends becomes a weighted average where the people who have more friends are weighted higher because they are observed more often. This phenomenon with weighted averages can also be seen in the class size paradox. Suppose a university offers only two course: one with 90 students and one with 10 students. It would say that it’s average course size is 50 students. However, if we were to ask the students, 90 students would say that the class size is 90 and 10 students would say that the class size is 10. Therefore, the students would say that the average course size is 82.

There are many interesting and practical applications for the friendship paradox. In a study done at Harvard University, two network scientists conducted a study on the outbreak of a flu pandemic. They randomly selected a set of students and then asked each person to name a friend and created a second set from those people. The second set got sick on average 2 weeks earlier than the first set. This is presumably because the second set was more highly connected.

We can take advantage of this paradox to quickly find a more people who might be the center of a social network. Currently, we know the Page Rank algorithm can be applied to a social network to give the people nearest to the center of the graph. However, Page Rank can be difficult to calculate. We can take advantage of the friendship paradox to find the center without having to calculate Page Rank or analyzing the entire social network. For example, if there were a limited number of vaccines in the above pandemic, we could require students to nominate a friend who would be given the flu shot. This would greatly increase the probability that the student receiving the flu shot is more centered in the social network.

-mso

### 2 Responses to “ The Friendship Paradox ”

• dwc92@cornell.edu

I had never heard of the friendship paradox before reading your post, and it was very interesting. I like how you point out that it’s not a sociological phenomenon, but actually an arithmetic theorem simply found in society, especially social networks. I didn’t fully understand the paradox until you used the class size paradox, which helped me realize what was happening with the weights. Your point about finding the center of a social network is interesting, although I don’t know how feasible it is to assume that people would rather nominate a friend for a flu shot rather than just take the flu shot themselves. Overall, well done!

• kyk25@cornell.edu

This blog post has a lot of basis in the Information Networks / World Wide Web section from the Networks I course — ie. the PageRank algorithm. As we learned, calculating the number of outlinks in comparison to the number of inlinks generates a “reputation” that gets used continuously to formulate a more accurate representation of each site. It’s interesting how this can also be utilized in a social network rather than an information network, with friends in place of links. If the model were given even more specificity, ie. outlinks would be people who you know of, but aren’t necessarily friends with (ie. celebrities, a leader on campus, etc), and inlinks are people who know of you, then there could be even more links created. This would in turn make the weights more accurate for the calculations, as the links aren’t necessarily mutual.

Overall, I found this to be an extremely interesting paradox and I enjoyed how the PageRank algorithm was used to describe a different type of network!