The friendship paradox: on average, your friends have more friends than you do. This is a mathematical fact about networks, not a psychological illusion. High-degree nodes appear in more people’s neighbor lists, inflating the average neighbor degree above the average node degree. It holds for any network with non-zero degree variance.

Except when it doesn’t (arXiv:2603.16337). Under degree-biased sampling — where the probability of observing a node is proportional to its degree — the expected degree of sampled vertices equals the expected degree of their neighbors. The paradox dissolves. Not because the network changed, but because the sampling method compensated for exactly the bias that created the paradox.

The result is mathematically equivalent to two other statements: the existence of a stationary random walk on the graph, and the conservation of total flow derived from vertex degree differences. The paradox, the stationary distribution, and the flow conservation are the same theorem in different clothing.

The through-claim: the friendship paradox was never a property of the network. It was a property of the sampling method. Uniform sampling over-counts high-degree nodes in the neighbor list without over-counting them in the node list, creating the asymmetry. Degree-biased sampling over-counts them in both lists equally, and the asymmetry cancels. The paradox lives in the gap between two different ways of weighting the same graph — and disappears when the weights align.