Pick a person at random and count their friends; then pick a random friend and count theirs — the friend almost always wins. Popular people show up on many friend-lists, so a random friend is over-sampled toward the well-connected. It’s pure size-biasWhen you sample proportional to size — picking a random friend rather than a random person — larger values appear more often than their frequency in the population. This 'size bias' inflates every estimate you draw from the sample., and the formula is exact: friends’ average = μ + σ²/μ.
You have 8 friends on average; your friends average 12.5 — about 56% more. The whole gap is variance: σ²/μ = 4.5. If everyone were equally connected (sd 0), it would vanish.
When sd = 0 the curve touches the reference line — no spread, no gap. As the network grows more unequal, friends’ average climbs away from the mean along a parabola (μ + σ²/μ). The vertical marker shows your current sd.
| CVCV = σ/μ, the coefficient of variation. It measures relative inequality: CV 0 is perfectly equal, CV 1 means spread equals the mean, CV 2 means very high inequality. (σ/μ) | Friends’ average | % more |
|---|---|---|
| 0 | 8 | 0% |
| 0.25 | 8.5 | 6.3% |
| 0.5 | 10 | 25% |
| 1 | 16 | 100% |
| 2 | 40 | 400% |
It’s the same size-biasSize-bias (also called length-biased sampling): when you pick by traversing an edge rather than a vertex, you are more likely to land on high-degree nodes — they have more edges. The same effect makes bus wait times feel longer than the timetable says. as the inspection paradox — popular people appear on many friend-lists, so a random friend skews high-degree. The gap is exactly σ²/μ, so the more unequal the network, the wider it grows; in a perfectly equal network it disappears. Flipped around, it’s why monitoring random people’s friends gives cheap early warning of an outbreak: the friends are over-sampled toward the highly connected, so they catch a spreading pathogen sooner.