Generalized Friendship Paradoxes in Network Science

TL;DR

This work develops a comprehensive linear-algebraic framework for generalized friendship paradoxes in undirected networks, distinguishing global and local averaging and applying it to a broad class of centralities. It proves universal global and local paradoxes for Katz-style and nonbacktracking centralities, and shows a reverse paradox for loneliness defined by reciprocal degree, including a global geometric mean variant. The results rely on spectral decompositions, nonbacktracking matrix analysis, and zeta-function connections, yielding precise equality conditions (often regularity or cycles) and highlighting that global and local versions are not equivalent. Practically, these findings inform sampling strategies and monitoring of information or disease spread, by clarifying when neighbors are expected to exhibit higher attribute levels across multiple centrality notions.

Abstract

Generalized friendship paradoxes occur when, on average, our friends have more of some attribute than us. These paradoxes are relevant to many aspects of human interaction, notably in social science and epidemiology. Here, we derive new theoretical results concerning the inevitability of a paradox arising, using a linear algebra perspective. Following the seminal 1991 work of Scott L. Feld, we consider two distinct ways to measure and compare averages, which may be regarded as global and local. For global averaging, we show that a generalized friendship paradox holds for a large family of walk-based centralities, including Katz centrality and total subgraph communicability, and also for nonbacktracking eigenvector centrality. Defining loneliness as the reciprocal of the number of friends, we show that for this attribute the generalized friendship paradox always holds in reverse. In this sense, we are always more lonely, on average, than our friends. We also derive a global averaging paradox result for the case where the arithmetic mean is replaced by the geometric mean. The nonbacktracking eigenvector centrality paradox is also established for the case of local averaging.

Theorems & Definitions (19)

• Theorem 2.1: Global friendship paradox Feld91
• Theorem 2.2: Global eigenvector centrality paradox H19
• Theorem 2.3: Local friendship paradox CKN21
• Theorem 2.4: Generalized local friendship paradox for eigenvector, walk and Katz centralities HV25
• Theorem 3.1: Generalized global friendship paradox for $f(A)\mathbf{1}$
• proof
• Corollary 3.1: Examples arising from $f(A)\mathbf{1}$
• Theorem 3.2: Generalized global nonbacktracking eigenvector centrality paradox
• proof
• Theorem 4.1: Generalized local nonbacktracking eigenvector centrality paradox