Friendship Paradox (part 2)

Last week I introduced the Friendship Paradox and confirmed it through a random simulation which illustrates its validity.

It's worth noting that the Friendship Paradox suggests that, on average, the number of friends we have is less than or equal to the number of friends our friends possess. This is indeed accurate! While it may seem counterintuitive, this can be demonstrated through mathematical proof. Still, I understand why it might feel contradictory to our everyday experiences.

Only if everyone had the same number of friends would the average number of friends per individual align with the average number of friends that their friends have. In all other cases, however we analyze it, it typically appears our friends have a higher friend count than us.

Consider a straightforward scenario involving three individuals.

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

The table below illustrates the number of friends each person has, alongside the average number of friends that their friends hold. For instance, Alice is friends with Charlie, giving her one friend. Similarly, Bob is also friends with Charlie, resulting in the same count. Conversely, Charlie has two friends: Alice and Bob. When calculating the averages, we find that 1.33 for direct friends is less than the average of friends of friends, which is 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 typical user boasts around 249 friends. In contrast, the average number of friends that their friends possess is 359 (source: Zach Star)

While I maintain the broader claim that, on average, our friends possess greater popularity than we do, the figure of 1.67 in the bottom right corner of the table may not be the most accurate focus. Statistically, it's problematic to derive an average from other averages.

What if we assessed how many friends each of our friends has and calculated the average from that data? In this scenario, Alice would note that her friend Charlie has 2 friends, contributing 2 points to Charlie. Bob would provide the same feedback, adding another 2 points to Charlie's score. Charlie then sees that Alice has 1 friend while Bob also has 1 friend. Compiling these scores totals to 6 points for Charlie, with Alice and Bob each having 1 point. This results in a total of 6 points shared among 4 distinct friendships. Thus, the average number of points per friendship is 6 divided by 4, resulting in 1.5, which still exceeds the average of 1.33 per individual.

What underlies this phenomenon in general? One potential explanation is that charismatic individuals tend to attract more friendship requests than those who are socially awkward. It’s common for most individuals to have a network filled with outgoing personalities rather than solitary ones.

In simpler terms, a small group of charismatic friends is elevating the average number of friends that our friends have. As we often befriend these outgoing individuals, it raises the average number of friends of our friends, leading to feelings of social awkwardness or exclusion among us.

Although there is a wealth of scholarly articles addressing this issue, I believe the essence boils down to the straightforward explanation provided here.