Monday, October 25, 2010

Your Friends Are More Likely To Have More Friends Than You Do

Have you ever wondered if your friends might be more popular than you are? Well, statistically it has some merits. On average, your friends will have more friends than you do. Scott L. Feld, Professor of Sociology at Purdue University, wrote about this counter-intuitive fact back in 1991.

This means if you were to ask number of friends your friends have, their average will be more likely be greater than number of friends that you have. Why would this be?

Top: Alice, Bob, and Carl are friends
Bottom Left: Dan becomes friend with Bob, most likely scenario
Bottom Right: Dan can become friend with Carl; unlikely configuration
Let's think about a simple case with three friends, Alice, Bob and Carl. The most interesting case is illustrated on the top: Alice and Bob are both friends of Carl. In this example, Alice's friends have more friends on average because Bob has two friends and Carl has one, averaging out to be 1.5 friends, and this is greater than number of Alice's friend, 1. Note that this is also true for Carl by symmetry. Only Bob bucks this trend.

Let's add Dan to this picture. Dan does not know any one from the group, Alice, Bob, and Carl, and gets invited to the cocktail party where Alice, Bob and Carl are also invited by third party. If you look at probability of Dan becoming a friend of Alice, Bob, or Carl, it's one third chance for each. But chances are Bob will be the most gregarious out of the three, and you can argue that there will be higher chance for Bob to befriend Dan.

This case is illustrated in lower left. Dan becomes a friend of Bob, and you can see how earlier trend continues. Alice, Carl, and Dan will have 1 friend each while their friends have average of (1 + 1 + 3) / 3 = 1.66 friends. Again Bob is the only exception.

What if Dan becomes a friend with Carl instead (illustrated on lower right)? Well, in that case we have a tie. While Alice and Dan have 1 friend each, their friends have 1.66 friends, so they continue on the trend of your friend having more friends. But Bob and Carl have 2 friends each, even though their friends have 1.33 friends on average, which is against the trend.

It is, however, extremely unlikely for this configuration to continue where people are only connected to only two friends to form a topological line. In real life people become friends through mutual friends. The graph become more densely connected with people intermingling with extended friends. While doing so, some friends retain the connection because of mutual interest or commonality more so than others.

Your friends are more likely to have more friends that you do because those popular friends (connectors) appear often in your friendship connection. In second example, Bob's three friends are counted three times because he has three friends. More popular you are, more likely you'll have friends, and that will skew the average friends for your friends. Satoshi Kanazawa expanded this even further to say a few things about popularity of your spouse on average on Psychology Today.

So don't despair. It's only natural that your friends look more popular than you are. Because most likely they are on average.


  1. I never thought of it that way. However, it is just fine with me because I really don't take these things seriously.

  2. That's certainly one approach. As for me, I've decided building massive number of follower or friends is not the game that I will play. Well, at least not for now.