Every friendship has two ends.
Popular people sit at the end of many friendships, so they turn up on far more friend lists than their numbers suggest.
Ask everyone to name their friends. Each name lands on the person named.
The gap is the spread.
If everyone had the same number of friends, there would be no paradox. Unevenness is the whole effect.
Where the formula comes from
Why the spread is measured against the average
Try to break it.
You can build a crowd where most people have more friends than their friends. You cannot make the averages come out the other way.
Tap empty space to add a person. Drag from one person to another to make or end a friendship. Double-tap a person to remove them.
Why the averages can't lose
How you meet people decides who you meet.
Find people through their friends and you oversample the popular. Build any crowd, sample it, and watch the bias, then remove it.
How you meet people
Edit the crowd the same way as above: tap to add, drag to connect, double-tap to remove.
In real crowds.
A 1950s high school, a karate club, a circle of Facebook friends. Same result.
Everyone seems to be going.
Anything the popular do looks more common than it is. Here, a few people are going to a show.
Tap a person to send them or keep them home. Drag from one person to another to make or end a friendship.
Who goes
Where the illusion comes from
Outnumbered is the normal view.
From inside a network, people are seen through their friendships, and the well-connected are seen most often. Feeling outnumbered by your friends is a fact about the sample, not about you.
The same arithmetic runs wherever things are met in proportion to their size: the class a student sits in, the gap between buses a rider waits through. The inspection paradox is the same story on a subway platform.