Friendship Paradox — how many friends do your friends actually have?

Dr. R. Düsing · Osnabrück University
Friendship Paradox — Help
The claim

"Your friends have more friends than you do, on average." Sounds like self-doubt, but it's a mathematical fact for almost every social network (Feld 1991). Click a person in the tool: usually, their friends have more friends on average than they do themselves.

Why?

Especially well-connected people appear on many friend lists — you yourself appear on only as many lists as you have friends. So if you look at "people's friends," these highly connected people keep showing up and pull the average up.

The math

Average number of friends of a person: E[k]. Average number of friends when counting over friendships:

E[friend] = E[k²]/E[k] = E[k] + Var(k)/E[k]

Because Var(k) ≥ 0, E[friend] ≥ E[person] — equal only if everyone has the same number of friends. The more unequal the network (bigger differences in friend counts ⇒ bigger Var), the bigger the gap. Slide "connectivity" up and watch the difference.

Size-biased sampling

The core is size-biased counting: counting over friendships weights every person by their friend count. Same pattern as the inspection paradox (you land in long intervals) or "courses feel bigger than average."

Everyday life & usefulness

"Your followers have more followers." The gym always looks packed. Useful: "vaccinate/monitor random friends of random people" hits more central nodes on average — a cheap strategy without global network knowledge.

References

Feld, S. L. (1991). Why your friends have more friends than you do. American Journal of Sociology, 96(6), 1464–1477.

📋 A Simple Question (Scott Feld, 1991)
Comparing your own friend count with your friends' — who has more, on average? The answer turns out surprisingly systematic for almost every network. Below, make your prediction first, then click through the network and check.
① Your Prediction
Picture a random person. Do their friends have more, about the same, or fewer friends than they do themselves, on average?
② Click a Person
each node = one person · lines = friendships · node size = number of friends · click a node
Click a person in the network (or roll the dice) — and compare their friend count to their friends' average.
③ Tally Across All People
"Friends have more" holds for
—
—
Mean friends per person
—
E[k] = 2·edges/people
Mean friends of a friend
—
E[k²]/E[k]
Difference
—
= Var(k)/E[k]
Make a prediction first (① above), then click through the network.
Concepts
Why "more"?
If you ask random people, each has the same chance of being picked. If instead you go over friendships ("people's friends"), every person gets counted as many times as they have friends. Well-connected people appear on many lists and thus keep showing up — they pull the "friends" average up. That's why almost everyone feels their friends are more popular than they are.
The formula
E[friend] = E[k²]/E[k] = E[k] + Var(k)/E[k] ≥ E[k]. Equality only if everyone has the same number of friends. More inequality ⇒ bigger gap.
Size-biased sampling
Counting over friendships weights every person by their friend count. Same principle as the inspection paradox and class-size bias.
What's it useful for?
"Monitor random friends of random people" hits more central nodes on average — a cheap early-detection/vaccination strategy without global network knowledge. → related effects
🦾 Fun Fact: Almost Everyone Has "Above-Average Arms"
Practically every human has 2 arms — a small few have only one or none due to accident or birth. The average is thus just under 2 (e.g. 1.999). Result: over 99% of people have more arms than average. In the friendship paradox, it's exactly the opposite: there, most people have fewer friends than "the average of their friends."

A similar aha-moment, but a different mechanism: in the arms example, it's purely due to the skew of the distribution (median ≠ mean). The friendship paradox additionally arises from size-biased sampling over edges — well-connected people get counted multiple times.