What is the friendship paradox?

The friendship paradox is the observation, first made by the sociologist Scott L. Feld in 1991, that an individual's friends typically have more friends than the individual does. It is called a paradox because it runs against what most people believe: most people think they have more friends than their friends have. It is not a statement about any one person's worth or likability. It is a property of how networks are built. [1]

The usual explanation is sampling bias. People with many friends appear on many friend lists, while people with very few friends appear on few. When you look at your own friends, you are therefore more likely to land on well-connected people, and less likely to land on people with almost no connections. [1] [2]

A PBS News explainer describes a double effect. A small group of hyperconnected people, such as celebrities with more than 100 million followers, pull up the average popularity of friends for everyone linked to them. Although such people are rare, they also show up in an extraordinary number of social circles. [3]

In network terms, each person is a node, and the number of friends is the node's degree. A paper in the Journal of Complex Networks states that the average degree of a node's neighbours is strictly greater than the average degree of the network as a whole. The proof shows that the average gap between a node's neighbours' degrees and its own degree can be written as a sum of squared terms, so it cannot be negative. [4]

The gap is exactly zero only when everyone connected to each other has the same number of friends, which is a regular graph. In every other network the effect is strictly positive on average. Because the result concerns averages, a given person can still be more popular than their friends, and the PBS explainer notes that an exceptionally gregarious person will not experience the paradox. [4] [3]

Averages can mislead. The Journal of Complex Networks paper gives a contrived example of a 1000-node network, a complete graph with one edge removed. There, 998 nodes are more popular than the average of their neighbours, yet two outliers skew the average so that the paradox still holds overall. Researchers have therefore studied the full distribution of differences. A Santa Fe Institute summary of this work says popular people tend to be friends with one another, and less popular people with one another, and that the wide range from a few friends to hundreds magnifies the effect. It reports that around 95% of the variation in the networks studied can be explained by these two features. [4] [5]

A stronger version also holds for many people: most of your friends have more friends than you do, not just the average. The PBS piece also cites a study finding that 98 percent of Twitter users follow accounts with more followers than they have. A generalized friendship paradox covers other traits, such as citation counts among collaborators, whenever the trait correlates with how connected someone is. [3] [4]