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

TL;DR

This paper analytically relates two common formulations of the friendship paradox, connecting their difference to degree-degree covariance and unifying node-level and moment-level perspectives.

Contribution

It establishes the exact relationship between alter-based and ego-based means of the friendship paradox, linking covariance and moments of the degree distribution.

Findings

The difference between the two formulations depends on degree-degree covariance.

Explicit examples cover assortative, neutral, and disassortative mixing patterns.

The covariance form and moment-based expression are shown to be equivalent.

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 [Proc. Natl. Acad. Sci. 121, e2306412121 (2024)], which involves the (-1)st, 1st, 2nd, and 3rd 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. The analysis thus unifies node-level and moment-level perspectives on the friendship paradox, offering both a pedagogically transparent derivation and a direct bridge to recent theoretical developments.