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Friendship paradox
You will learn: Count people and friendship endpoints separately to explain different average degrees.
Start with: Expectation and variance
Imagine six people: one knows the other five, and each of those five knows only the central person. A uniformly selected person has an average of 10/6 friends. But a name selected uniformly from everyone’s friend lists has an average of three friends. The same network gives different averages because the second experiment counts popular people repeatedly.
The familiar phrase “your friends have more friends than you do” compresses several possible comparisons. Here we separate a population identity from each person’s local comparison. We also distinguish choosing a random friendship endpoint from choosing a random person and then one of their friends.
Inspect the network and three sampling rules
Person 1: degree 5; friends 2, 3, 4, 5, 6; mean degree of these friends 1.000000. The selected person has a filled, thicker outline.
| Sampling experiment | Expected degree of sampled person |
|---|---|
| Uniform person | 1.666667 |
| Uniform directed edge, take its endpoint | 3.000000 |
| Uniform nonisolated person, then uniform friend | 4.333333 |
Degree variance 2.222222. 5 of 6 nonisolated people have fewer friends than their own friends average. Isolated people: 0; their local friend average is undefined.
Inspect degrees and sampling weights
| Person | Degree | Uniform-person weight | Endpoint weight | Local friend mean |
|---|---|---|---|---|
| 1 | 5 | 0.166667 | 0.500000 | 1.000000 |
| 2 | 1 | 0.166667 | 0.100000 | 5.000000 |
| 3 | 1 | 0.166667 | 0.100000 | 5.000000 |
| 4 | 1 | 0.166667 | 0.100000 | 5.000000 |
| 5 | 1 | 0.166667 | 0.100000 | 5.000000 |
| 6 | 1 | 0.166667 | 0.100000 | 5.000000 |
Each line is a mutual friendship, and the degree of a person is the number of incident lines. The graph has no self-friendships or repeated links. All nodes have equal visual size; the numbers identify people rather than ranks. The random preset independently includes each possible edge with the chosen probability. These are constructed networks, not measurements of real relationships.
Start with the six-person star. Inspect person one, whose degree is five and whose friends’ mean degree is one. Then inspect a leaf, whose degree is one and whose only friend has degree five. Five people lie below their local friend average; the central person lies above it. The population effect does not hold for every individual.
Count the appearances
Write dᵢ for person i’s degree. If there are n people, a uniform person has expected degree μ = Σdᵢ/n. A uniformly selected directed edge ends at person i with probability dᵢ/Σdⱼ, because that person appears once per incoming friendship. Its expected endpoint degree is therefore Σdᵢ²/Σdᵢ.
Let the degree variance over uniformly selected people be σ² = Σ(dᵢ − μ)²/n. Expanding the square gives Σdᵢ²/n = μ² + σ². When at least one friendship exists, dividing by μ yields endpoint mean μ + σ²/μ. Variance is nonnegative, so the endpoint mean cannot be smaller than the uniform-person mean. Equality holds when all degrees are equal.
For the star, the degrees are 5, 1, 1, 1, 1, 1. Their sum is ten and their squared sum is thirty. The central person receives half the endpoint weight, while each leaf receives one tenth. The endpoint average is 30/10 = 3. Choosing a uniform person gives the center only one sixth of the weight and produces 10/6 instead.
A random person’s random friend is another experiment
Now choose one of the six people uniformly and then choose one of that person’s friends uniformly. Five of the six starting people are leaves, so five sixths of the time the selected friend is the center, with degree five. Starting at the center selects a leaf with degree one. The expectation is (5/6)5 + (1/6)1 = 26/6, not three.
In a general graph, this protocol averages the local friend means across nonisolated starting people. The endpoint protocol instead averages across directed edges. People with many friends contribute more edges, but only one starting-person slot. The table deliberately shows both outcomes rather than naming either one “the average friend” without specifying how the friend was selected.
Make a prediction
In a complete network, must most people still have fewer friends than their friends?
Explore the answer
No. Every person has the same degree, so every local friend average equals that degree. The endpoint identity has zero variance and becomes equality.
Boundaries and interpretation
An isolated person contributes degree zero to the uniform-person mean but has no friend to choose. The person-then-friend rule explicitly excludes isolated starting people, and the local comparison reports its eligible denominator. If the entire graph is empty, both friend-sampling expectations are undefined; they are not zero-valued observations.
The algebra explains a sampling effect. It does not establish that a given person is unpopular, that every social network has the same shape, or that a numerical average captures relationship quality. Scott Feld’s original paper develops the friendship paradox. Continue to the inspection paradox to see the corresponding difference between counting intervals and sampling time.