Friendship Paradox (part 2)

Last week I introduced the Friendship Paradox and quantified it with evidence from a random simulation that it is in fact true.

 

As a reminder, the Friendship Paradox says that, on average, our friends have an equal or greater number of friends than we do. This is true! There is nothing paradoxical about it as it can be mathematically proven. However, I agree that it seems to violate common sense.

Only in the situation where everybody has the same number of friends will the average number of friends per person be equal to the average number of friends of their friends. Otherwise, however we measure it, it will appear our friends have more friends that we do.

Let’s look at a simple example of three people.

Alice is friends with Charlie.
Bob is friends with Charlie.
Charlie is friends with Alice and Bob.

The following table shows the number of friends per person as well as the average number of friends of their friends. For example, Alice has one friend (Charlie). Bob also has one friend (Charlie). Charlie has two friends (Alice and Bob). The bottom row shows the average number of friends, 1.33, is less than the average of average friends of friends, 1.67.

 

Person	Friends	Average Friends of Friends
Alice	1	2
Bob	1	2
Charlie	2	1
Average	1.33	1.67

 

On Facebook the average person has 249 friends. Meanwhile, the average number of friends of their friends is 359 (source: Zach Star)

While I still hold the general statement that our friends are more popular than we are, on average, maybe the 1.67 figure in the lower right of the table is not what we should be looking at. Statistically speaking, it isn’t kosher to take an average of averages.

What if we counted how many friends each friend has and take the average of that? In this case, Alice would say her friend Charlie has 2 friends, giving Charlie 2 points. Bob would say the same thing, giving Charlie two more points. Charlie would say Alice has 1 friend and Bob has 1 friend. Adding the points, Charlie has 4, Alice has 1 and Bob has 1. That is a total of 6 points. Between the two sides of two friendships, there are 4 total friends. The average points per friend is 6/4 = 1.5. This is still more than the average number of friends per person of 1.33.

Why is this true in general? One way to explain is that most people are more likely to be approached for friendship by charismatic people than socially awkward people. For most people it is to be expected they know a lot of social butterflies and not many loners.

In other words, a minority of charismatic people are blowing up the average of how many friends our friends have. We are likely to be friends with these people and that inflates the average friends of our friends, making us feel socially awkward and left out.

There are lots of mathematical papers on this topic, but I think it boils down the simple explanation above.