Friendship Paradox

Feld showed how uniformly sampling friendship-tie endpoints overrepresents people with many ties. In an undirected network, their mean degree is at least the uniform-person mean; it is higher when degrees vary.

The mean-degree effect is mathematically established and observed in real networks. Under uniform edge-endpoint sampling, the difference for another attribute depends on its covariance with degree. Sampling a random neighbour of a uniformly chosen person is a different rule.

Choose an undirected edge uniformly and then one of its two endpoints uniformly: a node is sampled with probability proportional to its degree. This creates degree-weighted exposure relative to uniform node sampling.

AI transfer to investigate: AI retrieval through graph links, citations or popularity signals may make the same degree bias scalable.

Proposed test, not yet conducted: Compare uniform-node retrieval with degree-weighted graph retrieval on synthetic networks in which attributes do or do not correlate with degree. Measure prevalence estimates and expert discovery.

Sampling bias can sometimes serve a useful purpose. The error is not popularity weighting itself but forgetting which estimand it changes.

Degree is a node’s number of connections. With uniform edge-endpoint sampling, mean degree is at least the uniform-person mean, with equality when all degrees are equal. This is an average result, not a claim about each individual.

Feld (1991), Why Your Friends Have More Friends Than You Do — American Journal of Sociology 96:1464–1477; pp.1465, 1470–1473 (PDF pp.2,7–10)

Cantwell, Kirkley & Newman (2021), The friendship paradox in real and model networks; arXiv v1 (2020) — §II equations (3)–(8); §III.4 comparison across 32 real networks; §IV

Feld (1991, pp. 1470–1473) and Cantwell et al. (arXiv version 1, §II) explain the sampling rule and the average. The proposed application to AI retrieval has not been independently tested.