Two variants of the friendship paradox: The condition for inequality between them

Abstract

The friendship paradox—the observation that, on average, one’s friends have more friends than oneself—admits two common formulations depending on whether averaging is performed over edges or over nodes. These two definitions, the “alter-based” and “ego-based” means, are often treated as distinct but related quantities. This paper establishes their exact analytical relationship, showing that the difference between them is governed by the degree-degree covariance normalized by the mean degree. Explicit examples demonstrate the three possible cases of positive, zero, and negative covariance, corresponding respectively to assortative, neutral, and disassortative mixing patterns. The derivation further connects the covariance form to the moment-based expression introduced by Kumar, Krackhardt, and Feld (2024), which involves the moments of the degree distribution. The two formulations are shown to be equivalent, as they should be: the moment-based representation expands the same structural dependence that the covariance form expresses in its most compact and interpretable form.