The Friendship Paradox

by | May 7, 2019


I am intrigued by paradoxes and came across a very interesting one called the Friendship Paradox which states:

On average, your friends have more friends than you do.

How can this be true? Its a function of the attributes of social networks. According to the MIT Technology Review, “the paradox arises because numbers of friends people have are distributed in a way that follows a power law rather than an ordinary linear relationship. So most people have a few friends while a small number of people have lots of friends.”

People with lots of friends are likely to be over-represented in groups of friends because people with lots of friends are in a lot of friend networks due to the fact they have a lot of friends. Mathematically, when one of these people with lots of friends is one of your friends they greatly increase the average number of friends that your friends have, resulting in your friends on average having more friends than you.

Related IFOD: How Many Friends Can You Really Have?

A good analogy is as follows: “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.”

A Diagram of a Friend Network: The person represented by the dark circle has six friends, denoted by gray circles. His friends have nine friends on average. These people are represented by white circles. Over the entire network, twenty-nine of the thirty-four people have friends who are more popular than they are.
Source: Page, Scott E.. The Model Thinker

The friendship paradox was discovered in 1991 by Scott Feld in the “offline” real world and has been found to apply to the online world as well.

The friendship paradox has been confirmed to exist on Facebook via research at Cornell University. The Cornell study 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”.

Similarly, study from USC found that the friendship paradox is true for over 98% of Twitter users. Specifically, “everyone you follow or who follows you has more friends and followers than you.” Further, “your friends are more active than you, on average” and send and receive more viral content.

The friendship paradox has been found to have analogues in other types of networks, such as:

academic citation networks, email networks, sexual contact networks, banking networks, and international trade networks. On average, the references cited by an academic article receive more citations than the article itself; a country’s trading partners, on average, trade with more countries than the country itself; and the multiple species connected to a single species in a food network have, on average more connections than the single species itself.

Source: The Model Thinker, by Scott E. Page

One potential application of the friendship paradox is predicting the spread of disease, including pandemics. In an intriguing study 319 Harvard undergrads were chosen at random who then named 425 of their friends. The friendship paradox holds that the 425 named friends should have more friends than the randomly chosen undergrads (on average). Both the randomly chosen group and the named friends were sent periodic surveys about whether they felt flu-like symptoms.
Students in the friend group showed signs of the flu between 14 and 69 days before the epidemic peaked in the control group of randomly selected undergraduates.” This study is promising in terms of being able predict and control epidemics with knowledge of social/friendship networks. According to a Professor at USC: “if you arrive in an African village with only five Ebola vaccines, the best strategy is not to vaccinate five random people, but ask those people who their friends are and vaccinate five of these friends. Due to the friendship paradox, the friends are likely to be more central, both in the Twitterverse and in the village, and thus more likely get sickened early by the virus or to tweet about topics that later become popular.”

1 Comment

  1. This paradox leaves a lot to think about! Interesting John!


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