Welcome to the Friendship Paradox

Remember what it was like in junior high? That’s when life became a lot more complicated. Low self-esteem… Negative body image… Recognizing the development of a social hierarchy that unfairly elevated those with upper body strength and early development of facial hair… And, of course, the depressing realization that everyone had more friends than you did.

Thankfully, that season of life is finite, and if you just wait long enough, those complexities go away. On top of that, your perception of being unpopular is largely a figment of your imagination. At least, that’s what Mom told us. Several decades later, we’re beginning to suspect that Mom was secretly mocking us and feeding us a line to make us stop whining.

Here we are, thirty years later, and some of us have yet to outgrow “the awkward years.” On top of that, we now know that we weren’t imagining things when we suspected our friends have more friends than we do.

The phenomenon is known as the Friendship Paradox. It is described in detail by Scott L. Feld in his 1991 paper “Why Your Friends Have More Friends Than You Do.” (American Journal of Sociology, 96(6), 1464–1477). Reading it reminds us that all of our friends our better at math than we are, too. For those who need some help understanding the explanation, consider it this way:

Imagine that the entire world consists of eight people: Alex, Betty, Charlie, Daniel, Erica, Freda, Gerry, and Harry. As shown in the diagram, Alex and Erica are clearly the extroverts of the world. Each of them has five friends. Curiously, they are not friends with each other, and neither of them is friends with Charlie. Charlie has only one friend, and that is Freda, who continually stresses to him that they are, in fact, “just friends.” Freda is also friends with Alex and Erica, giving her a total of three pals. Betty, Harry, and Gerry are each friends with Alex and Erica, but no one else, so each of them has two friends.

Now suppose you were to ask each of them how many friends they have, they would respond with, “Hey, who are you? We thought there were only eight people on the planet!” After they get over their shock, however, the answers would be as follows:

• Alex’s friends: 5
• Betty’s friends: 2
• Charlie’s friends: 1
• Daniel’s friends: 2
• Erica’s friends: 5
• Freda’s friends: 3
• Gerry’s friends: 2
• Harry’s friends: 2

Now suppose you ask each of these folks, “How many friends do your friends have?” This is how they would respond:

• Alex’s friends’ friends:
  • 5 people with a combined total of 11 friends, or an average of 2.2 friends per friend
• Betty’s friends’ friends:
  • 2 people with a combined total of 10 friends, or an average of 5 friends per friend
• Charlie’s friends’ friends:
  • 1 person who has three friends
• Daniel’s friend’s friends:
  • 2 people with a combined total of 10 friends, or an average of 5 friends per friend
• Erica:
  • 5 people with a combined total of 16 friends, or an average of 3.2 friends per friend
• Freda:
  • 2 people with a combined total of 10 friends, or an average of 5 friends per friend
• Gerry:
  • 2 people with a combined total of 10 friends, or an average of 5 friends per friend
• Harry:
  • 2 people with a combined total of 10 friends, or an average of 5 friends per friend

To put it another way, Alex has five friends, and his friends have, on average 2.2 friends each. Erica also has five friends. Her friends average 3.2 friends apiece. Clearly, Alex and Erica were the Homecoming King and Queen and never suffered from self-esteem issues.

The same cannot be said of anyone else on the planet, however. Betty, Daniel, Freda, Gerry, and Harry each have two friends, but their friends average 5 friends each. Bringing up the rear is Charlie, whose sole friend rubs it in his face that she has three friends.

Wait a minute… Is this some sort of statistical legerdemain? Friendships are supposed to be two-way relationships, right? Anyone who thinks otherwise is generally classified as a stalker, not a friend. Since friendships are bilateral, it seems that everything should all balance out.

That’s where Feld’s research helps us understand something that seems obvious once you see it. The fact is that you are more likely to be friends with someone who has more friends than you would be with someone who has fewer friends. In our example above, there are eight people. Each is counted once when counting the friends of an individual. When it comes to calculating friends of friends, each of the eight has the potential of being counted seven separate times. Alex and Erica, for example, each show up in the friends list of five people. Freda may think of Alex and Erica as her BFFs, but guess what? So do just about everyone else on the planet.

To say it another way, overly friendly people, if given a chance, are going to develop friendships with everyone they can find. If you are within reach, you’re probably going to get a friend request. You are not alone, though, because the same overly friendly person will be doing the same to your next-door neighbor. Conversely, an introverted person is not nearly as likely to reach out to you in friendship, nor will he or she be particularly enamored by your next-door neighbor.

The next time you are feeling bad about yourself because you don’t seem to be as popular as your friends, just remember that this is a normal, statistical result. There’s nothing to feel bad about.

Unless, of course, you are quite simply a horrible person and no one likes you. If that’s the case, no amount of statistical manipulation is going to help.

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