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

Sang Hoon Lee

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.

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 [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.

Paper Structure

This paper contains 11 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tale of Two FP Types
  3. Relation between the two FP types
  4. Examples
  5. $\langle k_{\textnormal{friend}}\rangle_\mathrm{n} = \langle k_{nn}\rangle_\mathrm{n}$ (neutral case, $\operatorname{Cov_\mathrm{n}}=0$)
  6. $\langle k_{\textnormal{friend}}\rangle_\mathrm{n} > \langle k_{nn}\rangle_\mathrm{n}$ (assortative case, $\operatorname{Cov_\mathrm{n}}>0$)
  7. $\langle k_{\textnormal{friend}}\rangle_\mathrm{n} < \langle k_{nn}\rangle_\mathrm{n}$ (disassortative case, $\operatorname{Cov_\mathrm{n}}<0$)
  8. Connecting the covariance and moment-based formulations
  9. Recap of the degree-moment formalism of Ref. Kumar2024
  10. Connecting the covariance forms and its implication
  11. Discussion and conclusions

Figures (1)

  • Figure 1: (a) A simple five-node network where the alter-based and ego-based means coincide (neutral case). The network values are: $\langle k\rangle_\mathrm{n} = 2$, $\langle k_{\text{friend}}\rangle_\mathrm{n} = 11/5$, $\langle k_{nn}\rangle_\mathrm{n} = 11/5$, and $\operatorname{Cov_\mathrm{n}}(k,k_{nn}) = 0$. (b) A network with mild assortativity: high-degree nodes tend to connect to one another. The network values are: $\langle k\rangle_\mathrm{n} = 7/3$, $\langle k_{\text{friend}}\rangle_\mathrm{n} = 18/7$, $\langle k_{nn}\rangle_\mathrm{n} = 5/2$, and $\operatorname{Cov_\mathrm{n}}(k,k_{nn}) = 1/6$. (c) A star-like configuration exhibiting strong disassortativity. The network values are: $\langle k\rangle_\mathrm{n} = 8/5$, $\langle k_{\text{friend}}\rangle_\mathrm{n} = 5/2$, $\langle k_{nn}\rangle_\mathrm{n} = 17/5$, and $\operatorname{Cov_\mathrm{n}}(k,k_{nn}) = -36/25$.