## The Friendship Paradox – Why your friends have more friends than you do

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

In his New York Times article, Cornell Applied Mathematics Professor Steve Strogatz uses mathematics to explain why your friends always seem to have more friends than you do. He references recent research about the Facebook social graph which found that “a user’s friend count was less than the average friend count of his or her friends, 93 percent of the time” and that “users had an average of 190 friends, while their friends averaged 635 friends of their own”. This may seem counter-intuitive, but it’s just like how a random person entering the gym might think that compared to themselves, the people working out at the gym are in better shape than they are. They’re probably right, because the kind of people who go to the gym are, well, the kind people who go to the gym. In other words, the gym is a biased sample. Similarly, the kind of people who you’re friends with are the kind of people who have friends. Your friends are a biased sample.

This relates to class when we considered a social network as a kind of graph where the nodes are people and the edges are pairs of friendships. At the same time, your friends don’t always have more friends than you do. Take for example, this graph:

A has 3 friends whereas, his friend B only has 2 friends. So the Friendship Paradox doesn’t hold here. The reason is that the above graph is dense, while the friendship paradox only holds when the social graph is sparse. As we reasoned about in class, the Global Friendship Network is probably not fully connected because it just takes one person to be left out in order for the entire graph to be disconnected. For similar reasons, we can assume that the Global Friendship Network is sparse.

Strogatz also mentions how Network scientists Nicholas Christakis and James Fowler showed that you can predict the spread of H1H1 by utilizing the fact that the friends of randomly chosen people have more friends and are more central than the randomly chosen people themselves. This ideas is very relevant to Game Theory because the actions of people with the flu very obviously affect the actions of others. By utilizing Game theory and the Friendship Paradox, one may be able to predict the best way to organize people to optimize the healthiness of the population as a whole.

-PuzzleSimplex

References:

J. Ugander, B. Karrer, L. Backstrom and C. Marlow, “The anatomy of the Facebook social graph.”

S. L. Feld, “Why your friends have more friends than you do,” American Journal of Sociology, Vol. 96, No. 6 (May 1991), pp. 1464–1477